# The Solution of Embedding Problems  in the Framework of GAPs with Applications  on Nonlinear PDEs

Authors: Reinhard Starkl
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

The paper presents a special class of embedding problems whoes solutions are important for the explicit solution of nonlinear partial differential equations. It is shown that these embedding problems are solvable and explicit solutions are given. Not only are the solutions new but also the mathematical framework of their
construction which is defined by a nonstandard function theory built over nonstandard algebraical structures, denoted as “GAPs”. These GAPs must not be neither associative nor division algebras, but the corresponding function theories built over them preserve the most important symmetries from the classical complex function theory in a generalized form: a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.


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## Body

## 1. Introduction

Except some small areas, at time the nonlinear world is inaccessible for analytic methods, that is, for methods without using numerical algorithms, we only know some explicit solutions of a few nonlinear differential equations (see e.g., [1–13]), and we know precious few about the “embedding’’ of nonlinear structures without any symmetry in higher dimensional structures with symmetry ([14–16], etc.).This fact legitimates the creation of a new mathematical theory, presented in   [17], which allows an “analytic access’’ on wider regions of nonlinearity. It was shown that this theory succeeds in solving nonlinear partial differential equations (concretely the Einstein equations from General Relativity) if the problem defining parameters (concretely the stress energy tensor) have a certain symmetry which can be seen as a wide generalization of the classical symmetry of holomorphy. This new symmetry—in the following denoted as “*-symmetry’’ or “GAP-symmetry’’ allows the explicit solution of Einstein equations and also of a broad variety of other partial differential equations.This leads to the following question as considered here: is it possible to embed an arbitraryn-dimensional nonlinear partial differential equation (which only symmetry is given by smoothness of all coefficient functions) in an N-dimensional nonlinear partial differential equation with *-symmetry, N≥n. If this holds, we can solve the given n-dimensional system by embedding this system into the N-dimensional system, which is solvable on the base of *-symmetry. Since embedding of an equation means embedding of the coefficient functions (which usually are given by vector fields (tensor fields)), our question above boils down to the following question:is it possible to embed an arbitraryn-dimensional smooth vector field  (tensor field) into a “*-symmetrical’’N-dimensional tensor field,N≥n?This question will be answered positively here.For a better understanding of the practical meaning of this result, some analogies to the classical situation of holomorphy are given (see the following section and also more details in   [17]).(i)
The simplest GAP is defined by the two-dimensional fieldℂ of the complex numbers. In this case, the *-symmetry is defined by the classical symmetry of holomorphy. It is a well-known fact, that an arbitrary smooth real-valued function f:I→ℝ, I⊆ℝ, can be “embedded’’ in a complex valued function F:G→ℂ, G⊆ℂ in the sense that the real part of F for values in the region I is identically with f. Our Embedding Theorem is a generalization of this classical result, by generalizing the two-dimensional field ℂ to an n-dimensional GAP (generally a nonassociative, noncommtative, nondivision algebra).(ii)
The classical symmetry of holomorphy leads immediately to related (partly equivalent) symmetries: the well-known Cauchy-Riemannian differential equations and the well-known Cauchy Integral Theorem. The GAP-formalism generalizes these symmetries: it is shown that in GAPs, analogies of the Cauchy-Riemannian differential equations exist (in the following denoted as Pseudo Cauchy-Riemannian differential equations or shorter: “PCRE’’) as well as analogies of the Cauchy Integral Theorem (in the following denoted as Pseudo Cauchy Integral Theorem or shorter: “PCIT’’).(iii)
The classical symmetries: Cauchy-Riemannian differential equations and Cauchy’s Integral Theorem lead to a solution theory of the two-dimensional linear Laplace equation. In a similar way, their analogies—the PCRE and the PCIT—lead to a solution theory ofn-dimensional nonlinear partial differential equations.After these considerations, some remarks to the structure of the paper are presented as follows.(1)
In Section2.1 the concept of “*-analyticity’’ is introduced: we will define the so-called *-analytic tensor fields by demanding that these fields allow a representation as generalized power series in very general algebras.(2)
In Section2.2 we will specialize these algebras to the so-called “PAk-structures’’ by introducing symmetries which can be seen as wide generalizations/modifications of the classical associativity symmetry. In the function theories built up over special PAk-structures (so-called “pseudorings’’), there exist generalizations of the well-known Cauchy-Riemannian differential equations as well as generalizations of the well-known Cauchy Integral Theorem.(3)
In Section2.3 a further generalization is given by generalizing the PAk-structures to the so-called “GAPs.’’ It is shown that also in this wide framework the main results of Section 2.2 hold.(4)
Section2.4 solves the problem of calculating GAPs explicitly.(5)
Section3 presents applications of the GAP-formalism on famous partial differential equations  from Theoretical Physics: new explicit solutions of Einstein equations  from General Relativity and Navier-Stokes equations  are given. The request of a further generalization of these results (which would be important from a physical point of view) shows the necessity of “embedding low dimensional unsymmetrical structures into higher dimensional symmetrical structures,’’ that is, the necessity of an Embedding formalism, which is developed in the following Section 4.(6)
Section4 presents the Embedding Theorem: it is shown that an arbitrary n-dimensional smooth vector field always can be embedded into a special N-dimensional smooth vector field (N≥n) restricted by special GAP-symmetries. In other words, The world of GAPs is wide enough to allow embedding for rather general (smooth) structures. It is remarkable that the Embedding Theorem not only shows the possibility of Embedding but also endows the tools for practical applications. A simple example is given.Finally some remarks to the style of the presented paper, which is given by accentuation of constructiveness, are presented like it is demanded by the concrete problems: not only are statements to existence and uniqueness desired and given but furthermore the explicit construction of a wide variety of solutions. For this reasons, this paper is not written for pure mathematicians but for physicists and applied mathematicians.


## 2. Elements of GAP-Theory

In this chapter the most important mathematical concepts will be presented as necessary for our solution method. Some of these concepts have been presented in   [17], where the interested reader can find the proofs as missing here. The larger part of concepts is new; the corresponding propositions of course all will be proved in the following.


### 2.1. The Concept of*-Analyticity in General Algebras

We start by remembering on same elementary terms of algebra theory: letV denote an arbitrary vector space built over the field ℝ of real numbers, and *:V×V→V an arbitrary binary distributive operation. The algebraic structure defined by V and * will be denoted here as (V;*). The *-operation on V-vectors can be described by means of an arbitrary V-base {bi}, i=1,…, dimV according to bi*bj=C(*)ijkbk, where C(*)ijk∈ℝ denote the structure constants  of the algebra (V;*) with respect to the base {bi}. Instead of i=1,…,dimV, we will write in the future i∈I1,dimV. The associated base-independent object to the structure constants C(*)ijk, i,j,k∈I1,dimV is the structure constant tensorC(*) of the algebra: C(*)=C(*)ijkbi⊗bj⊗bk, with bi∈V*, where V* denotes the dual space  of V and {bi} the dual base of {bj}, defined by bi(bj)=δji. By means of C(*) the product x*y,x,y∈V can be developed as follows:
(2.1)x*y=(x*y)kbk=(xibi)*(yjbj)=distributivityxiyj(bi*bj)=definitionofC(*)xiyjC(*)ijkbk⇒(x*y)k=xiC(*)ijkyj,k∈I1,dimV.
In the future we will denote the quadratic matrix xiC(*)ijk by (x*)jk and the quadratic matrix C(*)ijkyj by (*y)ik or shorter in formal denotation  x* and *y. If rankx*<dimV for a fixed element x, then equation x*y=0 has a nonzero solution y∈V and if rank*y<dimV for a fixed element y, then equation x*y=0 has a nonzero solution x∈V. We remember that in  the special case of a ring operation * a nonzero element x with x*y=0,y another nonzero element, is called a zero divisor, and a ring without zero divisors is called a division ring. In this work we will use the denotation “zero divisor’’ also for more general algebras in the sense above.If(V;*) has a left unit element (shorter: “left unit”), we denote this element as e(*,L), defined by e(*,L)*x=x,forallx∈V. If (V;*) has a right unit element (shorter: “right unit”), we denote this element as e(*,R), defined by x*e(*,R)=x,forallx∈V. If left unit and right unit are identical we will write e(*). The existence of a right unit or a left unit allows to define the following generalizations of the inverse element conception:
(2.2)x*x-1(*,R,R)=e(*,R),x-1(*,L,R)*x=e(*,R),x*x-1(*,R,L)=e(*,L),x-1(*,L,L)*x=e(*,L).
Here the symbol x-1(*,R,R) denotes the r̲ight inverse of x in respect of the right unit e(*,R) (see the superscripts R,R), x-1(*,L,R) denotes the l̲eft inverse of x in respect of the r̲ight unit e(*,R) (see the superscripts L,R), and so forth. The existence of a unit element e(*)=e(*,R)=e(*,L) allows the definition of an inverse element x-1(*) for x∈V: x-1(*)*x=x*x-1(*)=e(*). The extension (*) in x-1(*) will help us to distinguish an inverse vector from the inverse A-1 of a quadratic matrix A.The classical algebra symmetries (anti)commutativity, associativity, Jacobi-symmetry, existence of a left unit/right unit can be described by means of the structure constants as shown in Table1.Table 1SymmetryDefinitionCorrespondingC(*)-symmetry(anti)commutativityx*y=(-)y*xC(*)ijk=(-)C(*)jikassociativity(x*y)*z=x*(y*z)C(*)ijlC(*)lkm=C(*)jklC(*)ilmJacobi symmetry(x*y)*z+(y*z)*xC(*)ijlC(*)lkm+C(*)jklC(*)lim+(z*x)*y=0+C(*)kilC(*)ljm=0e(*,L)-existencee(*,L)*x=xe(*,L)iC(*)ijk=δjke(*,R)-existencex*e(*,R)=xC(*)ijke(*,R)j=δjkTheC(*) symmetries follow immediately from representing the vectors x,y,z in respect of a vector base {bi} and by calculating the base products bi*bj,(bi*bj)*bk, and so forth, by the structure constants.LetVp:=V⊗V⋯⊗V (p times) denote the p-repeated tensor product of the vector space V and Vq:=V*⊗V*⋯⊗V* (q times) the q-repeated tensor product of the dual space V*. With bi1⊗bi2⋯⊗bip∈Vp, we denote an arbitrary Vp-base, and with bi1⊗bi2⋯⊗biq∈Vq the corresponding dual base. Then a tensor T∈Vp (also called as “tensor of type (p,0)’’) has a representation T=Ti1,…,ipbi1⊗bi2⋯⊗bip with Ti1,…,ip=T(bi1,bi2,…,bip) and a tensor S∈Vq (also called as “tensor of type (0,q)’’) has a representation S=Si1,…,iqbi1⊗bi2⋯⊗biq with Si1,…,iq=S(bi1,bi2,…,biq).After these well-known assumptions, we will start with GAP-Theory by introducing two new product operationsVp×V→Vp and Vq×V→Vq as follows.Definition 2.1.
Given an arbitrary algebra(V;*) with structure constant tensor C(*). Then the*-associatedproduct of orderp:⊛(*):Vp×V→Vp is defined by
(2.3)(bi1⊗bi2⋯⊗bip)⊛(*)bj:=C(*)i1jkbk⊗bi2⋯⊗bip+bi1⊗C(*)i2jkbk⊗bi3⋯⊗bip+⋯+bi1⊗bi2⋯⊗bip-1⊗C(*)ipjkbk,
and the*-dual associated product of order q: ⊛(*)dual:Vq×V→Vq by
(2.4)(bi1⊗bi2⋯⊗biq)⊛(*)dualbj:=C(*)kji1bk⊗bi2⋯⊗biq+bi1⊗C(*)kji2bk⊗bi3⋯⊗biq+⋯+bi1⊗⋯⊗biq-1⊗C(*)kjiqbk.It is necessary to write⊛(*) instead of ⊛, because later we will deal with different product operations *k, k=1,2,… and with the corresponding *k-associated operations ⊛(*k). In the case p=1, it holds by definition that ⊛(*)=*; in the case q=1, we write also *dual for ⊛(*)dual. Due to the linearity of the vector spaces Vp and Vq, the ⊛-products of the base vectors uniquely define products of the form T⊛(*)z and S⊛(*)dualz for arbitrary tensors T∈Vp, S∈Vq, and z∈V. For this we look at the following results:Proposition 2.2.
The components of productsT⊛(*)z,S⊛dualz, T∈Vp, S∈Vq, and z∈V are given by
(2.5)(T⊛(*)z)i1i2⋯ip=(Tki2⋯ipC(*)kji1+Ti1ki3⋯ipC(*)kji2+⋯+Ti1i2⋯ip-1kC(*)kjip)zj,(S⊛(*)dualz)i1i2⋯iq=(Ski2⋯iqC(*)i1jk+Si1ki3⋯iqC(*)i2jk+⋯+Si1i2⋯iq-1kC(*)iqjk)zj.
The following algebraical symmetries hold:(2.6a)(T1+T2)⊛(*)z=(T1⊛(*)z)+(T2⊛(*)z),(2.6b)(T1⊗T2)⊛(*)z=(T1⊛(*)z)⊗T2+T1⊗(T2⊛(*)z),(2.6c)(anti)symmetryofT⇔(anti)symmetryofT⊛(*)z,(2.6d)T⊛(*)e(*,R)=pT,e(*,R) the right unit of the operation *,T1,T2,T∈Vp, z∈V. Analog relations hold for the dual *-associated products.Definition 2.3.
We define the mapsP(*)[α]:Vp×V→Vp and P(*)dual[α]:Vq×V→Vq as follows:
(2.7)P(*)[α](A,z):=(⋯((A⊛(*)z)⊛(*)z)⋯⊛(*)z)⊛(*)z,α-times,P(*)dual[α](B,z):=(⋯((B⊛(*)dualz)⊛(*)dualz)⋯⊛(*)dualz)⊛(*)dualz,α-times,A∈Vp and B∈Vq arbitrary constants, z∈V. The map P(*)[α](A,z) is denoted as the *-generalized power function of the vector z, P(*)dual[α](B,z) as thedual*-generalized power function.From this definition, it follows in particular thatP(*)[1](A,z)=A⊛(*)z, P(*)[2](A,z)=(A⊛(*)z)⊛(*)z≠A⊛(*)(z*z), P(*)[3](A,z)=((A⊛(*)z)⊛(*)z)⊛(*)z, which is different from A⊛(*)((z*z)*z) as well as from (A⊛(*)(z*z))⊛(*)z, and so forth. Furthermore, it follows that
(2.8)P(*)[α+1](A,z)=P(*)[α](A,z)⊛(*)z,P(*)dual[α+1](B,z)=P(*)dual[α](B,z)⊛(*)dualz,P(*)[α](z,z)=(⋯((z*z)*z)⋯*z)*z=:z[α+1],P(*)[α](e(*,L),z)=(⋯((e(*,L)*z)*z)⋯*z)*z=z[α],
where we have introduced the denotation z[α] for reason of simplicity as well as to differ this expression from the components zα, that is, the square bracket for the “order index’’ α refers to the generalized power operation.By means of the generalized power functions, we will introduce now the concept of*-analytic tensor fields. For the following 𝔗qp(W), W⊆ℝn denotes the vector space of tensors of type (p,q) which are smooth in a region W.Definition 2.4.
Given an arbitrary algebra(ℝn;*) and arbitrary tensors T∈𝔗p(W), S∈𝔗q(W), we denote T and S, respectively, as *-analytic in region W if there exists an element z0∈W, which allows the following representations of T, respectively, S:
(2.9)Ti1⋯ip(z)={A[0]+∑α≥1P(*)[α](A[α],z-z0)}i1⋯ip={A[0]+A[1]⊛(*)(z-z0)+(A[2]⊛(*)(z-z0))⊛(*)(z-z0)+((A[3]⊛(*)(z-z0))⊛(*)(z-z0))⊛(*)(z-z0)+⋯}i1⋯ip,(2.10)Si1⋯iq(z)={B[0]+∑α≥1P(*)dual[α](B[α],z-z0)}i1⋯iq={B[0]+B[1]⊛(*)dual(z-z0)+(B[2]⊛(*)dual(z-z0))⊛(*)dual(z-z0)+((B[3]⊛(*)dual(z-z0))⊛(*)dual(z-z0))⊛(*)dual(z-z0)+⋯}i1⋯iq,z∈W, A[α] constants of type (p,0), B[α] constants of type (0,q). The set of all in region W*-analytic tensors T and S for a fixed z0 is denoted as 𝔗p(W;*)(z0) and 𝔗q(W;*)(z0).In other words,*-analytic tensor fields are Taylor series in respect of the (dual) *-generalized power function, that is, the Taylor coefficients must take a special place within the “chain structure’’ of the series. If the series above are infinite, we need a criterion for convergence. Of course convergence depends on the region W as well as on the concretely used algebra (ℝn;*). For a simple convergence criterion, we introduce in the space (ℝn)p the following norm. (Since  the vector space ℝn is self-dual, that is, ℝn=(ℝn)*, we can identify (ℝn)p with (ℝn)p=((ℝn)*)p, and the following norm also is a norm for elements Y∈(ℝn)p with ∥Y∥:=:∑i1i2⋯ip|Yi1i2⋯ip|,p≥1)
(2.11)∥X∥:=:∑i1i2⋯ip|Xi1i2⋯ip|,X∈(ℝn)p,p≥1.Proposition 2.5.
Given the norm (2.11) and the denotations of  from above, then the series (2.9) is absolutely convergent in the regionW⊂ℝnif
(2.12)|C(*)kjl|≤Ml,∀j,k∈I1,n,with∑l=1nMl≤1p,∑α≥0∥A[α]∥∥z-z0∥α<∞,z,z0∈W.
The series (2.10) is absolutly convergent in the region W⊂ℝn if
(2.13)|C(*)kjl|≤Nk,∀j,l∈I1,n,with∑k=1nNk≤1q,∑α≥0∥B[α]∥∥z-z0∥α<∞,z,z0∈W.We sum up in the following.Until now two concepts have been introduced: the concept of*-associated Products⊛(*),⊛(*)dualand the concept of*-analytic tensor fields. We have seen  that the operations⊛(*),⊛(*)dualalways have some simple symmetries without restricting the underlying*-operation, and that specializing of*cannot induce the classical symmetries of associativity, commutativity, and Jacobi-symmetry for the operations⊛(*),⊛(*)dual.By means of the*-associated products, the so-called*-analytic tensor fields have been defined. These tensor fields always have symmetries without restricting the underlying*-operation, that is,*-analyticity is a stronger symmetry than smoothness.


### 2.2.*-Analyticity in PAk-Structures. Nonstandard Function Theory

Now we will define such symmetries of an algebra(ℝn;*), so that the corresponding *-analytical tensor fields become interesting “nonstandard symmetries”. In particular, we are interested in symmetries of the partial derivatives, the covariant derivatives, and the Lie-derivatives of *-analytical tensor fields.Definition 2.6.
An algebraic structure(ℝn;*1,*2) is called pseudoassociative of type k (PAk-structure), k = 1,2, if the operations *1,*2 satisfy the following commutation relation:
(2.14)k=1:(x*1y)*2z=(x*1z)*2y,k=2:x*1(y*2z)=z*1(y*2x),forallx,y,z∈ℝn. In the special case *1=*2=*, we denote these structures as pseudorings of typek.It is easy to see that a pseudoring of arbitrary type is associative if and only if it is commutative and a pseudoring with left unit is always commutative.Proposition 2.7.
An algebraic structure(ℝn;*1,*2)is a PAk-structure,  k=1,2, if and only if the structure constant tensors  C(*1) and  C(*2) satisfy the following commutation relation  (The following index symbol “” is used as an index commutator.): (2.15a)k=1:C(*1)ijkC(*2)klm=C(*1)ilkC(*2)kjm,(equivalentwithC(*1)i[jkC(*2)|k|l]m=0),(2.15b)k=2:C(*2)ijkC(*1)lkm=C(*2)ilkC(*1)jkm,(equivalentwithC(*2)i[jkC(*1)l]km=0.)This Proposition allows a characterization of PAk-structures by means of the corresponding structure constant tensors. The problem of the explicit construction of PAk- structure constant tensors will be solved later. Now we will analyze some relationships between the inverse of a matrix (written as()-1) and the (left/right) inverse elements of a “corresponding” vector. We will see that such relationships exist for matrices and vectors derived from PAk-structures.Proposition 2.8.
Given a PA1-structure𝔛=(ℝn;*1,*2) with left unit e(*2,L) and a PA2-structure 𝔜=(ℝn;*1,*2) with right unit e(*1,R), then it holds for arbitrary vectors x∈ℝn:
(2.16)inalgebra𝔛:(x*1)-1=*2ywithx*1y=e(*2,L),inalgebra𝔜:(x*2)-1=y*1withx*2y=e(*1,R).In pseudorings the relations above take the very simple form:(2.17)Pseudoringoffirsttypewithleftunit:(x*)-1=*x-1(*,R,L),Pseudoringofsecondtypewithrightunit:(x*)-1=x-1(*,R,R)*,Commutativepseudoringwithunit:(x*)-1=*x-1(*).In the next step we will show that beside the algebraic characterization there also exists a geometrical characterization of PAk-structures. we will restrict here to a geometrical interpretation only of the pseudoringsymmetries of first type and look at a differentiable manifold𝔐 with an affine connection. As well-known, the corresponding curvature tensor Ω can be written in local coordinates as(2.18)Ωiljm=Γij,lm-Γil,jm+ΓijkΓklm-ΓilkΓkjm=Γi[j,l]m+Γi[jkΓ|k|l]m,
where Γimj denote the affine connection coefficients in local coordinates  (To avoid misunderstandings, we refer to the definitions and denotations given in the appendix: the curvature tensor of a manifold 𝔐 (or a tangent bundle) with a connection is denoted with Ω and its coefficients with Ωiljm, whereas in the more special case of a Riemannian manifold the curvature tensor is denoted as R and its coefficients with Riljm). On the right side, the second summand agrees formal with the left side of the relation (2.15a) for the simple case *1=*2, that is, for pseudorin gsymmetry of first type. This formal similarity is growing in the case of a flat manifold. In this case it holds Ωiljm=0 and if we choose a local coordinate system with the special symmetries (It can be shown, that coordinate systems with this symmetry exist and can be constructed explicitly) Γi[j,l]m=0, then the relation above reduces to Γi[jkΓ|k|l]m=0, which is formal identical with the symmetry (2.15a) of a pseudoring of first type. Therefore we seethat the curvature tensor prefers the symmetry of pseudoassociativity of first type and not the symmetry of associativity. (Associativity follows only in the case of flat spaces with torsion zero, that is, if Γimj=Γmij. In this case, Γimj shows the symmetry of the structure constant tensor of a commutative algebra and according to the considerations above it holds: commutativity + pseudoringsymmetries = commutativity + associativity).Since the pseudoring symmetry of first type seems geometric fundamental, we can expect very interesting symmetries of the corresponding*-analytic tensors. As remarked in the introduction of this section, we are interested especially in the partial derivatives, the covariant derivatives and the Lie derivatives of *-analytical tensor fields. All these derivatives can be written by means of the so called *-derivative, which will be defined now as follows.Definition 2.9.
GivenT∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0). If the following series expansions
(2.19)(𝔇(*)T)i1⋯ip(z):={A[1]+∑α≥2P(*)[α-1](αA[α],z-z0)}i1…ip={A[1]+2A[2]⊛(*)(z-z0)+(3A[3]⊛(*)(z-z0))⊛(*)(z-z0)+⋯}i1…ip,(𝔇(*)S)i1⋯iq(z):={B[1]+∑α≥2P(*)dual[α-1](αB[α],z-z0)}i1⋯iq={B[1]+2B[2]⊛(*)dual(z-z0)+(3B[3]⊛(*)dual(z-z0))⊛(*)dual(z-z0)+⋯}i1⋯iq
are convergent for z∈W, the uniquely defined elements 𝔇(*)T∈𝔗p(W;*)(z0) and 𝔇(*)S∈𝔗q(W;*)(z0) will be denoted as the *-derivative of tensorT and the *-derivative of tensorS.We also will write sometimesT′ instead of 𝔇(*)Tand analog for tensor S. From this definition and relation (2.6c) from Proposition 2.2, it follows immediately that(2.20)Tis(anti)symmetric⇔𝔇(*)Tis(anti)symmetric,Sis(anti)symmetric⇔𝔇(*)Sis(anti)symmetric.
Of course it is not ensured that this formal concept of *-derivative has the properties of the classical derivatives as to be a “direction-independent” limit of a difference quotient, and so forth. The concrete circumstances sensitive depend on the structure of the given algebra. To get interesting symmetries (we have to define what is interesting!) of the *-derivative, we will specialise the algebra to a pseudoring of first type.Theorem 2.10.
Given a pseudoring of first type(ℝn;*) and a tensor  T∈𝔗p(W;*)(z0),W⊆ℝn,p≥1. Then the partial derivatives of  T satisfy the symmetry
(2.21)T,mi1⋯ip=(T′)li2⋯ipC(*)lmi1+(T′)i1li3⋯ipC(*)lmi2+⋯+(T′)i1i2⋯ip-1lC(*)lmip.
If there exists a right unit  e(*,R) on  (ℝn;*) an inverse relation holds:
(2.22)(T′)i1⋯ip=1pT,mi1⋯ipe(*,R)m.Proposition 2.11.
Let(ℂ;*) denote the field of complex numbers. Then the relations (2.21) take the form of the classical Cauchy-Riemannian differential equations.Therefore, our considerations generalize the circumstances of classical complex function theory (and also those of modern function theory built over fields or division algebras) as follows:(2.23)fieldstructureofℂ→pseudoringstructureof(ℝn;*)dimension2ofℝ2→arbitrarydimensionnvectorfieldsonℝ2→tensorfieldsonℝn.
As the most remarkable generalization seems the generalization of the field structure to the structure of a nonassociative noncommutative pseudoring structur with zero divisors, these circumstances legitimize the denotation of (2.21) as Pseudo-Cauchy-Riemannian equations or shorter as PCRE. The analogon of Theorem 2.10  for tensors S∈𝔗q(W;*)(z0) is given as follows.Theorem 2.12.
Given a pseudoring of first type(ℝn;*) and a tensor  S∈𝔗q(W;*)(z0),W⊆ℝn,q≥1, then the partial derivatives of  S satisfy the symmetry (The following equations also will be denoted as PCRE.)
(2.24)Si1⋯iq,m=(S′)li2⋯iqC(*)i1ml+(S′)i1li3⋯iqC(*)i2ml+⋯+(S)i1i2⋯iq-1lC(*)iqml.
If there exists a right unit  e(*,R) of (ℝn;*), an inverse relation holds:
(2.25)(S′)i1⋯iq=1qSi1⋯iq,me(*,R)m.Now we will study the covariant derivative of*-analytical tensor fields. For this it is necessary to formalize the PCRE. By considering the properties of the *-associated products ⊛(*) and ⊛(*)dual from Proposition 2.2, we see the possibility of the following formulation:(2.26)∂T=𝔒(*)T⊛(*),∂S=𝔒(*)S⊛(*)dual.
This relations show that partial derivatives can be “transformed” onto “ordinary” *-derivatives. This will offer an interesting approach for solving partial differential equations by transforming these equations onto ordinary “*-differential equations”, that is, ordinary algebra-differential equations.For studying the covariant derivatives of*-analytical tensor fields we remember that for arbitrary tensors T∈𝔗p(W),S∈𝔗q(W),W⊂ℝn the covariant derivatives in component form are defined by(2.27)T;ji1i2⋯ip=T,ji1i2⋯ip+Tki2⋯ipΓkji1+Ti1ki3⋯ipΓkji2+⋯+Ti1i2⋯ip-1kΓkjip,Si1⋯iq;j=Si1⋯iq,j-Ski2⋯iqΓi1jk-Si1ki3⋯iqΓi2jk-⋯-Si1i2⋯iq-1kΓiqjk,
where Γjki denotes the Christoffel symbols of the chosen ℝn-coordinate system. Comparison with the relations of Proposition 2.2 shows a strong formal similarity which legitimates the following denotations:(2.28)(T⊛(Γ))i1i2⋯ip:=Tki2⋯ipΓkji1+Ti1ki3⋯ipΓkji2+⋯+Ti1i2⋯ip-1kΓkjip,(S⊛(Γ)dual)i1⋯iq:=Ski2⋯iqΓi1jk+Si1ki3⋯iqΓi2jk+⋯+Si1i2⋯iq-1kΓiqjk.Proposition 2.13.
Given a pseudoring of first type(ℝn;*)and tensors T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,p,q≥1, then the covariant derivatives  DT,DS satisfy the following symmetries:
(2.29)DT=𝔇(*)T⊛(*)+T⊛(Γ),DS=𝔇(*)S⊛(*)dual-S⊛(Γ)dual.Now we will study the Lie-derivatives of*-analytical tensor fields. For this we remember on the Lie-derivative 𝔏v:𝔗qp(W)→𝔗qp(W) of a tensor T∈𝔗qp(W) in respect to a vector field v:(2.30)(𝔏vT)j1⋯jqi1⋯ip=Tj1⋯jq,mi1⋯ipvm-Tj1⋯jqmi2⋯ipv,mi1-Tj1⋯jqi1mi3⋯ipv,mi2-⋯-Tj1⋯jqi1⋯ip-1mv,mip+Tmj2⋯jqi1⋯ipv,j1m+Tj1mj3⋯jqi1⋯ipv,j2m+⋯+Tj1⋯jq-1mi1⋯ipv,jqm.Proposition 2.14.
Given two pseudorings of first type(ℝn;*) and  (ℝn;∘), furthermore the tensors T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,p,q≥1and a vector v∈𝔗1(W;∘)(z0). Then the Lie-derivatives 𝔏vTand  𝔏vSsatisfy the following symmetries (The following operation  ∘T is defined by  C(∘T)mjk=C(∘)jmk, that is, by “transposition” of the corresponding quadratic matrix with fixed index  k.):
(2.31)𝔏vT=𝔇(*)T⊛(*)v-T⊛(∘T)𝔇(∘)v,𝔏vS=𝔇(*)S⊛(*)dualv+S⊛(∘T)dual𝔇(∘)v.
For a pseudoring of first type  (ℝn;*) with right unit  e(*,R) and a tensor field, it holds that (2.32a)𝔇(*)T=𝔏e(*,R)T,T∈𝔗p(W;*)(z0),(2.32b)(𝔏gf)′=𝔏gf′+𝔏g′f,f,v∈𝔗1(W;*)(z0).Relation (2.32a) shows a very simple characterization of the *-derivative by the Lie-derivative, but it is only possible in a pseudoring with a right unit.After the partial derivatives, covariant derivatives and Lie-derivatives we will study the exterior derivatives of*-analytic forms. Let us denote the set of all in region W⊆ℝn smooth q -forms with 𝔄q(W), and the subset of all in W*-analytical q-forms with 𝔄q(W;*)(z0).Proposition 2.15.
Given a pseudoring(ℝn;*) of first type, an arbitrary nontrivial q-form ω∈𝔄q(W;*)(z0),and arbitrary vector fields u,v∈𝔗1(W;*)(z0),w∈𝔗1(W;*)(z0),v not constant (“Not constant” means of course not constant in the considered coordinate system), then it holds that
(2.33)d⋀ω=0⇔(ℝn;*)iscommutative,u*v∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative,w*dualv∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative,u-1(*)∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative.This proposition shows that some fundamental symmetries with practical importance only exist in commutative pseudorings: the*-product of *-analytical vector fields is *-analytic only for commutative pseudorings, and an analogue result holds for the *dual-product. Also the inverse element of a *-analytical vector field will be *-analytical only in a commutative pseudoring. Let us sum up the relations for the classical derivatives as follows:Pseudo-Cauchy-Riemann equations:(2.34a)∂T=𝔇(*)T⊛(*),∂S=𝔇(*)S⊛(*)dual,
Covariant derivatives:(2.34b)DT=𝔇(*)T⊛(*)+T⊛(Γ),DS=𝔇(*)S⊛(*)dual-S⊛(Γ)dual,
Lie-derivative:(2.34c)𝔏vT=𝔇(*)T⊛(*)v-T⊛(∘T)𝔇(∘)v,𝔏vS=𝔇(*)S⊛(*)dualv+S⊛(∘T)dual𝔇(∘)v,𝔇(*)T=𝔏e(*,R)T,
Exterior  derivative:
(2.34d)d⋀ω=0⇔(ℝn;*)iscommutative,ω∈𝔄q(W;*)(z0),with (ℝn;*),(ℝn;∘) pseudorings of first type, T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,v∈𝔗1(W;∘)(z0). We sum up in the following paragraph.Our aim was to study*-analytical tensor fields in the framework of pseudorings, which have been introduced here as a fundamental concept. It was shown that pseudoringsymmetry is an algebraic symmetry which also appears in the world of differential geometry, moreover: the geometrical world prefers the pseudoringsymmetry against the classical symmetry of associativity. For this the study of pseudorings is legitimized, and so the study of tensor fields built over pseudorings. In particular we have studied the partial derivatives of*-analytical tensor fields, their covariant derivatives, their Lie-derivatives and their exterior derivatives, all this in the frame of pseudoringsymmetry. It was shown, that all these fundamental derivatives can be written in terms of the so-called*-derivative, which has been introduced here as a fundamental concept, following from the symmetry of*-analyticity.


### 2.3. Generalizations. The World of GAPs

Until now we have dealt with PAk-structures(ℝn;*1,*2) and pseudorings (ℝn;*), that is, with algebraic structures defined by maximal two product operations. For some later applications, this framework will be too small and shall be generalized here (The most applications on mathematical physics only need some small parts of this section: the concept of a GAP, of a GAP characteristic, and of GAP Exponentials). Such an algebra generalization will lead us to the concept of GAPs (PAk-structures and pseudorings will be shown as the most simple GAP-structures) and will allow us to generalize the function theoretical concepts of *-analyticity, *-derivative, *-integration, and so forth, from the section above by replacing the underlying PAk-structures (pseudorings) by GAPs.The aim of this section is to generalize the concepts of PAk-structures,*-analyticity,*-derivative,*-integration, Pseudo Cauchy Riemann equations, and so forth.Let us denote𝔛=(ℝn;*1,*2,…,*M) as an algebraic structure with the operations *α:ℝn×ℝn→ℝn,α∈I1,M,M an arbitrary natural number which also might be infinite. To generalize the concept of *-analyticity we have to generalize the concept of the power function from Section 2.1. For this we introduce the maps P(*1,*2,…,*α)[α]:(ℝn)p×(ℝn)→(ℝn)p and P(*1,*2,…,*α)dual[α]:(ℝn)q×(ℝn)→(ℝn)q as follows:(2.35)P(*1,*2,…,*α)[α](A,z):=(⋯((A⊛(*α)z)⊛(*α-1)z)⋯⊛(*2)z)⊛(*1)z,α-times,P(*1,*2,…,*α)dual[α](B,z):=(⋯((B⊛(*α)dualz)⊛(*α-1)dualz)⋯⊛(*2)dualz)⊛(*1)dualz,α-times,A∈(ℝn)p,B∈(ℝn)q arbitrary constants, z∈ℝn.  After these assumption we can introduce the concept of “chain-analyticity”.Definition 2.16.
Given an arbitrary algebraic structure𝔛=(ℝn;*1,*2,…,*M) and arbitrary tensors T∈𝔗p(W1),S∈𝔗q(W1), we denote T and S, respectively, as chain-analyticin region W1, if there exists an element z0∈W1, which allows the following representations of T, respectively, S:
(2.36)Ti1⋯ip(z)={A[0]+∑α=1MP(*1,*2,…,*α)[α](A[α],z-z0)}i1⋯ip:=A[0]+A[1]⊛(*1)(z-z0)+(A[2]⊛(*2)(z-z0))⊛(*1)(z-z0)+((A[3]⊛(*3)(z-z0))⊛(*2)(z-z0))⊛(*1)(z-z0)+⋯,Si1⋯iq(z)={B[0]+∑α=1MP(*1,*2,…,*α)dual[α](B[α],z-z0)}i1⋯iq:=B[0]+B[1]⊛(*1)dual(z-z0)+(B[2]⊛(*2)dual(z-z0))⊛(*1)dual(z-z0)+((B[3]⊛(*3)dual(z-z0))⊛(*2)dual(z-z0))⊛(*1)dual(z-z0)+⋯,z∈W1,A[α] a constant of type (p,0),B[α] a constant of type (0,q). The set of all in region W1 chain-analytical tensors T and S for fixed z0 is denoted as 𝔗p(W1;*1,*2,…,*M)(z0) and 𝔗q(W1;*1,*2,…,*M)(z0).Consider that in the series above an infinite value ofM is allowed and an infinite set of *α-operations is allowed to contain an infinite subset of identical operations. In the case *1=*2=⋯=*M  (=finite or infinite), we will denote the structures above as *-analytical as in the earlier case of a single product operation. The denotation “chain-analytical” comes from the fact, that the series above are built by *α-operations like the “elements of a chain”. We see, that for different *α-operations the structure of chain-analytical tensor fields is much more general than the structure of simple *-analytical tensor fields. Now we will generalize the concept of PAk-structures for vector spaces with different dimensions.Now I will present a concept, which allows a short overview about all symmetries—PAk-symmetry as well as standard symmetries—of a given algebraic structure. For this we assign the algebraic structure𝔛=(ℝn;*1,*2,…,*M) a M×M-matrix G(𝔛), denoted as the symmetry characteristic of the algebra𝔛 as follows:(2.37)G(𝔛)αβ:={kifthesubalgebra(ℝn;*α,*β)isaPAk-structure,0otherwise.}
(“0” means: “no PAk symmetry is assumed” and not: “no PAk-symmetry is allowed”. With other words: In the case “0” it is possible that a PAk-symmetry exists, but it is not ensured.)Definition 2.17.
An algebraic structure𝔛=(ℝn;*1,*2,…,*M)is called a GAP of order M, if the symmetry characteristic G(𝔛) includes at least one value 1 or 2. In this case G(𝔛) is denoted as the GAP characteristic of the GAP 𝔛.In other words a GAP is an algebraic structure which has at least one PA1-symmetry or one PA2 -symmetry. For further applications it will be advantageable, to generalize the GAP-characteristic by giving an overview not only about the PAk-symmetries but also about the ”standard symmetries” of all algebraic structures(ℝn;*α), that is, of its possible (anti)commutativity, Jacobi-symmetry, associativity, existence of a right unit e(*,R), existence of a left unit e(*,L), and so forth. Since these symmetries all are defined for a single algebra operation, they only will appear in the main diagonal of G(𝔛). Concretely we introduce the following denotations as subscripts of the elements G(𝔛)αα:(2.38)c~commutativity,ac~anticommutativity,J~Jacobi-symmetry,e(R)~existenceofarightunit,e(L)~existenceofaleftunit,e~existenceofaunit.
As an example, we consider an algebraic structure 𝔛=(ℝn;*1,*2,*3,*4) with the following symmetry characteristic:(2.39)G(𝔛)=(010000e(R)10000ac,J00200)
which means that the algebra (ℝn;*2) has a right unit, the algebra (ℝn;*3) is anticommutative and satisfies the Jacobi-symmetry (i.e., (ℝn;*3) is a Lie-algebra), the algebra (ℝn;*4) is commutative, the algebraic structure (ℝn;*1,*2) is a PA1-structure, and the algebraic structure (ℝn;*4,*2) is a PA2-structure. Per definition 𝔛 is a GAP.Now some special types of GAPs will be introduced which allow a wide generalization of the functional theoretical concepts of*-analyticity, *-derivative, Pseudo Cauchy Riemann equations, *-integration, and so forth.Definition 2.18.
A PA1-chain is a GAP𝔛=(ℝn;*1,*2,…,*M) with PAkαα-1=1,α∈I2,M. A PA1-chain is denoted as a closed PA1-chain, if additionally to the above it holds  PAk1M=1. A closed PA1-chain of Lie-typeis defined as a closed PA1-chain of order 3 with the additional restriction
(2.40)(x*3y)*2z+(y*2z)*1x+(z*1x)*3y=0,∀x,y,z∈ℝn.The concept of a PA1-chain allows a wide generalization of the earlier concept of the*-derivation, which will be presented now.Definition 2.19.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and chain-analytic tensors T∈𝔗p(W1;*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0), then the following series expansions
(2.41)(𝔇(*1)T)i1⋯ip(z):={A[1]+∑α≥2MP(*2,…,*α)[α-1](αA[α],z-z0)}i1⋯ip:={A[1]+2A[2]⊛(*2)(z-z0)+(3A[3]⊛(*3)(z-z0))⊛(*2)(z-z0)+⋯}i1⋯ip,(𝔇(*1)S)i1⋯iq(z):={B[1]+∑α≥2MP(*2,…,*α)dual[α-1](αB[α],z-z0)}i1⋯iq:={B[1]+2B[2]⊛(*2)dual(z-z0)+(3B[3]⊛(*3)dual(z-z0))⊛(*2)dual(z-z0)+⋯}i1⋯iq,z∈W1, uniquely define elements 𝔇(*1)T∈𝔗p(W1;*2,…,*M)(z0) and 𝔇(*1)S∈𝔗q(W1;*2,…,*M)(z0) which will be denoted as the *1-derivative of tensorT and the *1-derivative of tensorS.We see that the tensors𝔇(*1)T and 𝔇(*1)S do not contain the operation *1 any longer and so we cannot define 𝔇(*1)𝔇(*1)T or 𝔇(*1)𝔇(*1)S. Only 𝔇(*2)𝔇(*1)T and 𝔇(*2)𝔇(*1)S make sense, and also 𝔇(*3)𝔇(*2)𝔇(*1)T and 𝔇(*3)𝔇(*2)𝔇(*1)S, and so forth. Thus we will define the operators(2.42)𝔇(*k,*k-1,…,*1)k:=𝔇(*k)𝔇(*k-1)⋯𝔇(*1),k∈I1,M
as the chain derivatives of order k. In the special case *1=*2=⋯=*k:=* we write simplifying 𝔇(*)k instead of 𝔇(*,*,…,*)k. From the  definitions above  it follows for z∈W1:(2.43)(𝔇(*k,*k-1,…,*1)kT)i1⋯ip(z):={A[k]+∑α≥k+1MP(*2,…,*α)[α-k](α(α-1)⋯(α-k)A[α],z-z0)}i1⋯ip,(𝔇(*k,*k-1,…,*1)kS)i1⋯iq(z):={B[k]+∑α≥k+1MP(*2,…,*α)dual[α-k](α(α-1)⋯(α-k)B[α],z-z0)}i1⋯iq,
and we see that(2.44)𝔇(*k,*k-1,…,*1)kT∈𝔗p(W1;*k+1,…,*M)(z0),𝔇(*k,*k-1,…,*1)kS∈𝔗q(W1;*k+1,…,*M)(z0).Theorem 2.20.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and a chain-analytic tensorT∈𝔗p(W1;*1,…,*M)(z0),p≥1. Then the following symmetries hold:
(2.45)(𝔇(*k,*k-1,…,*1)kT),αi1⋯ip=(𝔇(*k+1,*k,…,*1)k+1T)li2⋯ipC(*k+1)lαi1+(𝔇(*k+1,*k,…,*1)k+1T)i1li3⋯ipC(*k+1)lαi2+⋯+(𝔇(*k+1,*k,…,*1)k+1T)i1i2⋯ip-1lC(*k+1)lαip,forallα∈I1,n1,i1,…,ip∈I1,n2,k∈I0,M-1 (In the case k=0 the left side must be interpreted as T,αi1⋯ip). An inverse relation exists, if the algebra (ℝn;*1) has a right unit  e(*1,R):
(2.46)(D(*k+1,*k,…,*1)k+1T)i1i2i3⋯ip=1p(𝔇(*k,*k-1,…,*1)kT),mi1⋯ipe(*1,R)m.Theorem 2.21.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and a chain-analytic tensor S∈𝔗q(W1;*1,…,*M)(z0),p≥1. Then the following symmetries hold:
(2.47)(𝔇(*k,*k-1,…,*1)kS)i1⋯iq,α=(𝔇(*k+1,*k,…,*1)k+1S)li2⋯iqC(*k+1)i1αl+(𝔇(*k+1,*k,…,*1)k+1S)i1li3⋯iqC(*k+1)i2αl+⋯+(𝔇(*k+1,*k,…,*1)k+1S)i1i2⋯iq-1lC(*k+1)iqαl,forallα∈I1,n1,i1,…,ip∈I1,n2,k∈I1,M-1 (In the case k=0 the left side must be interpreted as Si1⋯iq,α). An inverse relation exists, if the algebra (ℝn;*1) has a right unit  e(*1,R):
(2.48)(𝔇(*k+1,*k,…,*1)k+1S)i1i2⋯iq=1q(𝔇(*k,*k-1,…,*1)kS)i1⋯iq,me(*1,R)m.The proofs of Theorems2.20 and 2.21 run the same lines as the proofs of Theorems 2.10 and 2.12. Relations (2.45) and (2.47) will be denoted as Pseudo Cauchy Riemann equations (PCRE) as in the simple case of a single vector space and a single product operation. (The expression𝔇(*0,*0-1,…,*1)0T must be interpreted as T, and 𝔇(*0,*0-1,…,*1)0S as S.) Now we will generalize the former concept of *-integration.Definition 2.22.
Given an arbitrary algebra𝔛=(ℝn;*1,*2,…,*M)and chain-analytic tensorsT∈𝔗p(W1;*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0),p≥1, and further let (ℝn;*0) an algebra with the property
(2.49)(ℝn;*1,*0)isaPA1-structure,
Then the series
(2.50)(𝔇(*0)-1T)i1⋯ip(z):={∑α≥0MP(*0,*1,…,*α)[α+1](1α+1A[α],z-z0)}i1⋯ip+Ki1⋯ip:={K+A[0]⊛(*0)(z-z0)+(12A[1]⊛(*1)(z-z0))⊛(*0)(z-z0)+((13A[2]⊛(*2)(z-z0))⊛(*1)(z-z0))⊛(*0)(z-z0)+⋯}i1⋯ip,(𝔇(*0)-1S)i1⋯iq(z):={∑α≥0MP(*0,*1,…,*α)dual[α+1](1α+1B[α],z-z0)}i1⋯iq+Li1…iq:={L+B[0]⊛(*0)dual(z-z0)+(12B[1]⊛(*1)dual(z-z0))⊛(*0)dual(z-z0)+((13B[2]⊛(*2)dual(z-z0))⊛(*1)dual(z-z0))⊛(*0)dual(z-z0)+⋯}i1⋯ip,
with z∈W1,K and L arbitrary constants, are called the *0-integrals or the *0-antiderivatives of the tensors T and S.The denotation𝔇(*0)-1 is well chosen because it is 𝔇(*0)𝔇(*0)-1T=𝔇(*0)-1𝔇(*0)T and 𝔇(*0)𝔇(*0)-1S=𝔇(*0)-1𝔇(*0)S. We see, that(2.51)T∈𝔗p(W1;*1,…,*M)(z0)⇒𝔇(*0)-1T∈𝔗p(W1;*0,*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0)⇒𝔇(*0)-1S∈𝔗q(W1;*0,*1,…,*M)(z0).Proposition 2.23.
If𝔛is a PA1-chain, then the  *0-integrals  𝔇(*0)-1Tand 𝔇(*0)-1S can be expressed by the classical integral conception as follows:
(2.52)𝔇(*0)-1T(z)=∫T(z)⊛(*0)dz,𝔇(*0)-1S(z)=∫S(z)⊛(*0)dualdz.Theorem 2.24.
𝔛a PA1-chain and  𝔏 a smooth closed curve in region  W1. Then it is
(2.53)∮𝔏T(z)⊛(*0)dz=0,∮𝔏S(z)⊛(*0)dualdz=0.This Theorem generalizes the well-known classical Cauchy Integral Theorem for GAPs and is denoted asPseudo Cauchy Integral Theorem (PCIT). For the next step we remember on the classical denotation y′=dy/dx for a scalar differentiable function y(x), or formally y′dx=dy. It shall be shown now, that for the *-derivative of vector-valued functions a similar result holds:Proposition 2.25.
Given a PA1-chain𝔛=(ℝn;*0,*1,…,*M)and vectorsU∈𝔗1(W;*0,*1,…,*M)(z0),V∈𝔗1(W;*0,*1,…,*M)(z0). Then it holds
(2.54)dU(z)=𝔇(*0)U⊛(*0)dz,dV(z)=𝔇(*0)V⊛(*0)dualdz.Proof.
We start from the relation (2.52), with T∈𝔗1(W1;*1,…,*M)(z0) and S∈𝔗1(W1;*1,…,*M)(z0). Now we define U(z)=𝔇(*0)-1T(z),V(z)=𝔇(*0)-1S(z), from which follows U∈𝔗1(W;*0,*1,…,*M)(z0),V∈𝔗1(W;*0,*1,…,*M)(z0) as it is assumed above. Inserting U and V in (2.52) we obtain U(z)=∫𝔇(*0)U⊛(*0)dz,⇔dU(z)=𝔇(*0)U⊛(*0)dz, and analog:V(z)=∫𝔇(*0)V⊛(*0)dualdz,⇔dV(z)=𝔇(*0)V⊛(*0)dualdz.We sum up in the following paragraphIn this section, the main earlier results have been generalized,along with the algebraic structures as well as the tensor fields built over these structures. In particular the PAk-structures have been generalized by the concept of the so-called “GAPs”. We have investigated special GAPs like PA1 chains and their specializations (closed PA1-chains and closed PA1-chains of Lie-type), and have shown that the main results of the classical complex function theory (Cauchy Riemannian differential equations, Cauchy Integral Theorem) also hold in PA1-chains.


### 2.4. The Explicit Construction of GAPs

The aim of this section is the explicit construction of a wide variety of GAPs.In other words, we have to construct structure constant tensors of pseudorings, PAk-structures, and PA1-chains explicitly. This problem makes sense, because without knowledge of concrete GAPs, the results of the sections above would be not applicable in practice. In the following, we introduce the mapsHn:ℤ→]0,n], n∈ℕ (ℤ the set of integers, ℕ the set of natural numbers) with n≥1 by(2.55)Hn(x)={x-mn,ifmn<x≤(m+1)n,xif0<x≤n,x+mnif-mn<x≤(-m+1)n,
where m∈ℕ denotes the smallest possible value, defined uniquely by the variable x and the conditions above (For example, let us consider the case n=3,x=3.7. Then m is uniquely defined by m=1, since it is mn<x≤(m+1)n only for m=1. Then it is Hn(x)=x-mn=0.7). From this definition it follows immediately Hn(x+y)=Hn(y+x), and  Hn(x+Hn(y+z))=Hn(y+Hn(z+x))=Hn(z+Hn(x+y))=Hn(x+y+z).Proposition 2.26.
Given the constantsP∈ℕ,M∈I1,P,andAM∈ℝ,aM,σ∈ℤ.Then PAk-structures (ℝn;*1,*2) are given by the following structure constant tensors:(2.56a)casek=1:C(*1)ijk=∑M=1PAMδHn(aMi+σj)k,C(*2)klm=δHn(k+σl)m,(2.56b)casek=2:C(*1)ijk=δHn(σi+j)k,C(*2)klm=∑M=1PAMδHn(aMk+σl)m.Proof.
We show that the algebra (2.56a) satisfies the PA1-condition C(*1)i[jkC(*2)|k|l]m=0. Writing all sums explicitely we get ∑k=1nC(*1)ijkC(*2)klm=∑k=1n∑M=1PAMδHn(aMi+σj)kδHn(k+σl)m=∑M=1PAMδHn(Hn(aMi+σj)+σl)m, because δHn(aMi+j)k only delivers a nontrivial value for the index value k=Hn(aMi+j) and this index k must be inserted in the term δHn(k+σl)m. Since it is Hn(Hn(aMi+σj)+σl)=Hn(aMi+σj+σl), the right side above is symmetrical in indices j and l. Thus the left side is symmetrical in indices j and l, that is, the PA1-symmetry holds and statement (2.56a) has been proved. Now we have to show that algebra (2.56b) satisfies the PA2-condition C(*2)i[jkC(*1)l]km =0:∑k=1nC(*2)ijkC(*1)lkm =∑k=1n∑M=1PAMδHn(aMi+σj)kδHn(σl+k)m = ∑M=1PAMδHn(σl+Hn(aMi+σj))m = δHn(σl+aMi+σj)m. The right side is symmetrical in indices j and l, and so the left side is, that is, the PA2-symmetry holds and statement (2.56b) has been proved.Proposition 2.27.
Let ben∈ℕ,σ∈ℤarbitrary. Then the structure constant tensor
(2.57)C(*)ijk=δHn(i+σj)k,i,j,k∈I1,n
defines a pseudoring of first type  (ℝn;*). In the case  σ=1, a unit element exists, given by  e(*)j=δnj, and furthermore it hold s that(2.58a)∃λ∈ℝn,λ≠0with∄λ-1(*),(2.58b)Anonexistingλ-1(*)doesnotimplyλ2=0.Because the PAk-structures above are defined by Kronecker-symbols, we will denote them asPAk-structures of Kronecker-type or shorter asKronecker PAk-structures. According to the property (2.58a) a commutative Kronecker pseudoring is not a division algebra. It is easy to show, that this property also holds for noncommutative Kronecker pseudorings. Also the property (2.58a) can be generalized for noncommutative Kronecker pseudorings, where λ-1(*) must be replaced by λ-1(*,R,R), and so forth. (see the definition of λ-1(*,R,R),λ-1(*,R,L),λ-1(*,L,R),λ-1(*,L,L) in (2.2)). We see that a Kronecker algebra has a very simple structure, because the values of the structure constants are given by only two reals: 0 and 1. Now another type of PAk-structures will be presented which is more subtle than the Kronecker type.Proposition 2.28.
Given an algebra(ℝn;*1,*2) as follows:
(2.59)C(*α)ijh=diδjh+b(*α)djδih+didjh(*α)h,α=1,2,(informaldenotation:C(*α)=d⊗I+b(*α)I⊗d+d⊗d⊗h(*α))
with d,h(*1),h(*2)∈ℝn, d≠0,b(*1),b(*2)∈ℝ arbitrary nonzero constants. Then (ℝn;*1,*2) is a PAα-structure if and only if
(2.60)〈d,h(*α)〉=-1+σ(α){b(*2)-b(*1)σ(α)}withσ(α)=(-1)α-1,α=1,2.(ℝn;*α),α=1,2has a right unit  e(*α,R) if and only if
(2.61)〈d,h(*α)〉=-1,Thenitise(*α,R)=-b(*α)-1h(*α).The PAk-structures presented above are built by combinations of rank 2-tensors and rank 1-tensors. Therefore we will denote these structures asPAk-structures of splitting type or shorter as splitting PAk-structures.From this follows immediately the structure of splitting pseudorings. It is easy to show, that Kronecker PAk-structures generally cannot be transformed onto splitting PAk-structures by basistransformations in ℝn, that is, these both structures are really different.Splitting pseudorings have a large advantage to Kronecker pseudorings because it allow the explicit calculation ofzP,z-1, and so forth.Proposition 2.29.
Given the structure constant tensor(2.62)C(*)ijk=diδjk+bdjδik+didjhkwithhkdk=-1,(informaldenotation:C(*α)=d⊗I+bI⊗d+d⊗d⊗h,with〈h,d〉=-1)i,j,k∈I1,n,b∈ℝ only restricted by  b≠0.  Then the following statements hold:Statement 1.
The algebra(ℝn;*)defines ann-dimensional pseudoring of first type with right unite(*,R)=-b-1h.Statement 2.
In this algebra a vectorzhas a right invers elementz-1(*,R,R)in respect of the right unit,if and only if〈z,d〉≠0.Statement 3.
In(ℝn;*)the power functions, roots and inverses of vectors are calculable explicitly as follows:(2.63)z[P]=bP-2{(b+P-1)〈d,z〉P-1z+(P-1)〈d,z〉Ph},P∈ℕ,(2.64)z-1(*,R,R)=-b-2{b〈d,z〉-2z+(1+b)〈d,z〉-1h},
where z[P] is defined by z[P]=(⋯((z*z)*z)⋯)*z. In the case b=1 the relation (2.63) also holds for arbitrary rational numbers.Statement 4.
Forb=1it holds(2.65)Anonexistingλ-1(*)impliesλ2=0,U∈𝔗1(W;*)(z0)⇒λ<d,U>∈𝔗1(W;*)(z0)iff∄λ-1(*),U∈𝔗1(W;*)(z0)⇒d<λ,U>∈𝔗1(W;*)(z0)iff∄λ-1(*).Proposition2.29 shows, that in the framework of splitting pseudorings the inverse elements as well as the roots of vectors are calculable explicitly, that is, without using numerical methods. In Kronecker algebras this is not possible. Now we will show, that PA1-chains and furthermore PA1-chains of Lie-type can be constructed explicitly, proving, that the world of the PAk-structures is compatible with the world of generalized Lie-symmetries:Proposition 2.30.
The following algebraic structures define PA1-chains𝔛=(ℝn;*1,*2,…,*M):Kronecker PA1-chains:(2.66a)C(*α)ijk=δHn(i+σj)k,α=1,…,M,σ∈ℤarbitrary,
Splitting PA1-chains:
(2.66b)C(*α)ijk=diδjk+b(*α)djδik+didjh(*α)k,α∈I1,M,with1+b(*α)+〈d,h(*α)〉=b(*α-1),α∈I2,M.
Modified Splitting
(2.66c)C(*α)ijk=d(*α)iA(*α)jk+gjB(*α)ik,α∈I1,M,withtherestrictions
PA1-chains:
(2.66d)A(*α+1)jkd(*α)k=0,B(*α+1)jkd(*α)k=ud(*α+1)j,A(*α+1)jkB(*α)km=uA(*α)jm,u∈ℝarbitrary.Proposition 2.31.
The following algebraic structure(ℝn;*1,*2,*3) defines a closed PA1-chain from Lie type:
(2.67)C(*α)ijk=diδjk+b(*α)djδik+didjh(*α)k,α∈I1,3,withb(*1)=13(〈h(*2),d〉-〈h(*1),d〉),b(*2)=-1+23〈h(*2),d〉+13〈h(*1),d〉,b(*3)=1+〈h(*1),d〉,h(*1)mb(*2)+h(*2)mb(*3)+h(*3)mb(*1)=0.Until now we have delt with special PAk -structures (Kronecker structures and splitting structures). Now we will analyze the problem, if PAk-structures can be combined in a way, that we come to PAk-structures again. There are two ways to analyze this problem. The first way is to try a construction of newn-dimensional PAk-structure by combining n-dimensional PAk-structures, the second way is to try a construction of new N-dimensional PAk-structure by combining n-dimensional PAk-structures, N>n. We will give answers to both problems by the following both propositions.Proposition 2.32.
Given a GAP𝔛=(ℝn;*1,*2,*3,*4)with the characteristic
(2.68)G(𝔛)=(0010000100000000).
Then the algebra 𝔜=(ℝn;⊛1,⊛2)defined by
(2.69)C(⊛1)ijk:=C(*1)ijk+C(*2)ijk,C(⊛2)ijk:=C(*3)ijk+C(*4)ijk
is a PA1-structure, if and only if
(2.70)C(*2)i[jkC(*3)|k|l]m=-C(*1)i[jkC(*4)|k|l]m.
Given a GAP 𝔛=(ℝn;*1,*2,*3,*4)with the characteristic:
(2.71)G(𝔛)=(0100001000000000).
Then the algebra ℨ=(ℝn;⊚1,⊚2)defined by
(2.72)C(⊚1)ijk:=C(*1)ijsC(*2)spkup,C(⊚2)ijk:=C(*3)ijsC(*4)spkvp,u,v∈ℝnarbitrary
is a PA1-structure.Proof.
To prove the first statement we calculate(2.73)C(⊛1)ijkC(⊛2)klm=(C(*1)ijk+C(*2)ijk)(C(*3)klm+C(*4)klm)=C(*1)ijkC(*3)klm+C(*1)ijkC(*4)klm+C(*2)ijkC(*3)klm+C(*2)ijkC(*4)klm,⇒C(⊛1)i[jkC(⊛2)|k|l]m=C(*1)i[jkC(*3)|k|l]m+C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m+C(*2)i[jkC(*4)|k|l]m=GAP-symmetries(2.69)C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m.
We see, that the PA1 -symmetry C(⊛1)i[jkC(⊛2)|k|l]m=0 holds if and only if C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m=0, according to (2.70). To prove the second statement we calculate
(2.74)C(⊚1)ijkC(⊚2)klm=C(*1)ijsC(*2)spkupC(*3)klrC(*4)rqmvq,⇒C(⊚1)i[jkC(⊚2)|k|l]m=C(*1)i[jsC(*2)|spkupC(*3)k|l]rC(*4)rqmvq=GAP-symmetries(2.72)C(*1)i[jsC(*2)|s|l]kupC(*3)kprC(*4)rqmvq=GAP-symmetries(2.72)0.Proposition 2.33.
Given a PA1-structure𝔛=(ℝn;*1,*2). Then the following algebra 𝔜=(ℝ2n;⊛) is a pseudoring of first type:
(2.75)C(⊛)ijk={C(*1)ijk,ifi,j,k∈I1,n,C(*2)i-nj-nk-n,ifi,j,k∈In+1,2n,0,inallothercases.Proof.
Per defnition it is(2.76)C(⊛)ijkC(⊛)klm=∑k=1nC(⊛)ijkC(⊛)klm+∑k=n+12nC(⊛)ijkC(⊛)klm
for arbitrary indices i,j,l,m∈I1,2n. Let us consider first the case that i∈I1,n. Then nontrivial elements of C(⊛)ijk only exist in the case j,k∈I1,n, defined by C(⊛)ijk=C(*1)ijk, and the term above takes the form
(2.77)C(⊛)ijkC(⊛)klm=∑k=1nC(⊛)ijkC(⊛)klm=∑k=1nC(*1)ijkC(⊛)klm={0,ifl>norm>n,∑k=1nC(*1)ijkC(*2)klm,ifl,m∈I1,n.
Thus it follows for i∈I1,n:(2.78)C(⊛)i[jkC(⊛)|k|l]m={0,ifl>norm>n∑k=1nC(*1)i[jkC(*2)|k|l]m,ifl,m∈I1,n}=0
in every case, because the algebra (ℝn;*1,*2) is assumed as a PA1-structure, that is, it is ∑k=1nC(*1)i[jkC(*2)|k|l]m=0 for i∈I1,n.
Let us consider now the case thati∈In+1,2n. Then nontrivial elements of C(⊛)ijk only exist in the case j,k∈In+1,2n, defined by C(⊛)ijk=C(*2)i-nj-nk-n, and we get
(2.79)C(⊛)ijkC(⊛)klm=∑k=n+12nC(⊛)ijkC(⊛)klm=∑k=n+12nC(*1)i-nj-nk-nC(⊛)klm={0,ifl≤norm≤n,∑k=n+12nC(*1)i-nj-nk-nC(*2)k-nl-nm-n,ifl,m∈In+1,2n.
Thus it follows with the denotations ι̅=i-n,j̅=j-n,k̅=k-n,l̅=l-n,m̅=m-n:(2.80)C(⊛)i[jkC(⊛)|k|l]m={0ifl≤norm≤n,∑k̅=1nC(*1)ι̅[j̅k̅C(*2)|k̅|l̅]m̅inallothercases,}=0.
in every case, because the indices ι̅,j̅,k̅,l̅,m̅ move in the region I1,n, and the algebra (ℝn;*1,*2) is assumed as a PA1-structure.In this section a wide variety of GAPs has been constructed explicitly. In particular two different types of explicit calculable PAk-structures have been characterized: Kronecker PAk-structures and (generalized) Splitting PAk-structures. Also PA1-chains and even PA1-chains of Lie-type can be constructed explicitly.


## 2.1. The Concept of*-Analyticity in General Algebras

We start by remembering on same elementary terms of algebra theory: letV denote an arbitrary vector space built over the field ℝ of real numbers, and *:V×V→V an arbitrary binary distributive operation. The algebraic structure defined by V and * will be denoted here as (V;*). The *-operation on V-vectors can be described by means of an arbitrary V-base {bi}, i=1,…, dimV according to bi*bj=C(*)ijkbk, where C(*)ijk∈ℝ denote the structure constants  of the algebra (V;*) with respect to the base {bi}. Instead of i=1,…,dimV, we will write in the future i∈I1,dimV. The associated base-independent object to the structure constants C(*)ijk, i,j,k∈I1,dimV is the structure constant tensorC(*) of the algebra: C(*)=C(*)ijkbi⊗bj⊗bk, with bi∈V*, where V* denotes the dual space  of V and {bi} the dual base of {bj}, defined by bi(bj)=δji. By means of C(*) the product x*y,x,y∈V can be developed as follows:
(2.1)x*y=(x*y)kbk=(xibi)*(yjbj)=distributivityxiyj(bi*bj)=definitionofC(*)xiyjC(*)ijkbk⇒(x*y)k=xiC(*)ijkyj,k∈I1,dimV.
In the future we will denote the quadratic matrix xiC(*)ijk by (x*)jk and the quadratic matrix C(*)ijkyj by (*y)ik or shorter in formal denotation  x* and *y. If rankx*<dimV for a fixed element x, then equation x*y=0 has a nonzero solution y∈V and if rank*y<dimV for a fixed element y, then equation x*y=0 has a nonzero solution x∈V. We remember that in  the special case of a ring operation * a nonzero element x with x*y=0,y another nonzero element, is called a zero divisor, and a ring without zero divisors is called a division ring. In this work we will use the denotation “zero divisor’’ also for more general algebras in the sense above.If(V;*) has a left unit element (shorter: “left unit”), we denote this element as e(*,L), defined by e(*,L)*x=x,forallx∈V. If (V;*) has a right unit element (shorter: “right unit”), we denote this element as e(*,R), defined by x*e(*,R)=x,forallx∈V. If left unit and right unit are identical we will write e(*). The existence of a right unit or a left unit allows to define the following generalizations of the inverse element conception:
(2.2)x*x-1(*,R,R)=e(*,R),x-1(*,L,R)*x=e(*,R),x*x-1(*,R,L)=e(*,L),x-1(*,L,L)*x=e(*,L).
Here the symbol x-1(*,R,R) denotes the r̲ight inverse of x in respect of the right unit e(*,R) (see the superscripts R,R), x-1(*,L,R) denotes the l̲eft inverse of x in respect of the r̲ight unit e(*,R) (see the superscripts L,R), and so forth. The existence of a unit element e(*)=e(*,R)=e(*,L) allows the definition of an inverse element x-1(*) for x∈V: x-1(*)*x=x*x-1(*)=e(*). The extension (*) in x-1(*) will help us to distinguish an inverse vector from the inverse A-1 of a quadratic matrix A.The classical algebra symmetries (anti)commutativity, associativity, Jacobi-symmetry, existence of a left unit/right unit can be described by means of the structure constants as shown in Table1.Table 1SymmetryDefinitionCorrespondingC(*)-symmetry(anti)commutativityx*y=(-)y*xC(*)ijk=(-)C(*)jikassociativity(x*y)*z=x*(y*z)C(*)ijlC(*)lkm=C(*)jklC(*)ilmJacobi symmetry(x*y)*z+(y*z)*xC(*)ijlC(*)lkm+C(*)jklC(*)lim+(z*x)*y=0+C(*)kilC(*)ljm=0e(*,L)-existencee(*,L)*x=xe(*,L)iC(*)ijk=δjke(*,R)-existencex*e(*,R)=xC(*)ijke(*,R)j=δjkTheC(*) symmetries follow immediately from representing the vectors x,y,z in respect of a vector base {bi} and by calculating the base products bi*bj,(bi*bj)*bk, and so forth, by the structure constants.LetVp:=V⊗V⋯⊗V (p times) denote the p-repeated tensor product of the vector space V and Vq:=V*⊗V*⋯⊗V* (q times) the q-repeated tensor product of the dual space V*. With bi1⊗bi2⋯⊗bip∈Vp, we denote an arbitrary Vp-base, and with bi1⊗bi2⋯⊗biq∈Vq the corresponding dual base. Then a tensor T∈Vp (also called as “tensor of type (p,0)’’) has a representation T=Ti1,…,ipbi1⊗bi2⋯⊗bip with Ti1,…,ip=T(bi1,bi2,…,bip) and a tensor S∈Vq (also called as “tensor of type (0,q)’’) has a representation S=Si1,…,iqbi1⊗bi2⋯⊗biq with Si1,…,iq=S(bi1,bi2,…,biq).After these well-known assumptions, we will start with GAP-Theory by introducing two new product operationsVp×V→Vp and Vq×V→Vq as follows.Definition 2.1.
Given an arbitrary algebra(V;*) with structure constant tensor C(*). Then the*-associatedproduct of orderp:⊛(*):Vp×V→Vp is defined by
(2.3)(bi1⊗bi2⋯⊗bip)⊛(*)bj:=C(*)i1jkbk⊗bi2⋯⊗bip+bi1⊗C(*)i2jkbk⊗bi3⋯⊗bip+⋯+bi1⊗bi2⋯⊗bip-1⊗C(*)ipjkbk,
and the*-dual associated product of order q: ⊛(*)dual:Vq×V→Vq by
(2.4)(bi1⊗bi2⋯⊗biq)⊛(*)dualbj:=C(*)kji1bk⊗bi2⋯⊗biq+bi1⊗C(*)kji2bk⊗bi3⋯⊗biq+⋯+bi1⊗⋯⊗biq-1⊗C(*)kjiqbk.It is necessary to write⊛(*) instead of ⊛, because later we will deal with different product operations *k, k=1,2,… and with the corresponding *k-associated operations ⊛(*k). In the case p=1, it holds by definition that ⊛(*)=*; in the case q=1, we write also *dual for ⊛(*)dual. Due to the linearity of the vector spaces Vp and Vq, the ⊛-products of the base vectors uniquely define products of the form T⊛(*)z and S⊛(*)dualz for arbitrary tensors T∈Vp, S∈Vq, and z∈V. For this we look at the following results:Proposition 2.2.
The components of productsT⊛(*)z,S⊛dualz, T∈Vp, S∈Vq, and z∈V are given by
(2.5)(T⊛(*)z)i1i2⋯ip=(Tki2⋯ipC(*)kji1+Ti1ki3⋯ipC(*)kji2+⋯+Ti1i2⋯ip-1kC(*)kjip)zj,(S⊛(*)dualz)i1i2⋯iq=(Ski2⋯iqC(*)i1jk+Si1ki3⋯iqC(*)i2jk+⋯+Si1i2⋯iq-1kC(*)iqjk)zj.
The following algebraical symmetries hold:(2.6a)(T1+T2)⊛(*)z=(T1⊛(*)z)+(T2⊛(*)z),(2.6b)(T1⊗T2)⊛(*)z=(T1⊛(*)z)⊗T2+T1⊗(T2⊛(*)z),(2.6c)(anti)symmetryofT⇔(anti)symmetryofT⊛(*)z,(2.6d)T⊛(*)e(*,R)=pT,e(*,R) the right unit of the operation *,T1,T2,T∈Vp, z∈V. Analog relations hold for the dual *-associated products.Definition 2.3.
We define the mapsP(*)[α]:Vp×V→Vp and P(*)dual[α]:Vq×V→Vq as follows:
(2.7)P(*)[α](A,z):=(⋯((A⊛(*)z)⊛(*)z)⋯⊛(*)z)⊛(*)z,α-times,P(*)dual[α](B,z):=(⋯((B⊛(*)dualz)⊛(*)dualz)⋯⊛(*)dualz)⊛(*)dualz,α-times,A∈Vp and B∈Vq arbitrary constants, z∈V. The map P(*)[α](A,z) is denoted as the *-generalized power function of the vector z, P(*)dual[α](B,z) as thedual*-generalized power function.From this definition, it follows in particular thatP(*)[1](A,z)=A⊛(*)z, P(*)[2](A,z)=(A⊛(*)z)⊛(*)z≠A⊛(*)(z*z), P(*)[3](A,z)=((A⊛(*)z)⊛(*)z)⊛(*)z, which is different from A⊛(*)((z*z)*z) as well as from (A⊛(*)(z*z))⊛(*)z, and so forth. Furthermore, it follows that
(2.8)P(*)[α+1](A,z)=P(*)[α](A,z)⊛(*)z,P(*)dual[α+1](B,z)=P(*)dual[α](B,z)⊛(*)dualz,P(*)[α](z,z)=(⋯((z*z)*z)⋯*z)*z=:z[α+1],P(*)[α](e(*,L),z)=(⋯((e(*,L)*z)*z)⋯*z)*z=z[α],
where we have introduced the denotation z[α] for reason of simplicity as well as to differ this expression from the components zα, that is, the square bracket for the “order index’’ α refers to the generalized power operation.By means of the generalized power functions, we will introduce now the concept of*-analytic tensor fields. For the following 𝔗qp(W), W⊆ℝn denotes the vector space of tensors of type (p,q) which are smooth in a region W.Definition 2.4.
Given an arbitrary algebra(ℝn;*) and arbitrary tensors T∈𝔗p(W), S∈𝔗q(W), we denote T and S, respectively, as *-analytic in region W if there exists an element z0∈W, which allows the following representations of T, respectively, S:
(2.9)Ti1⋯ip(z)={A[0]+∑α≥1P(*)[α](A[α],z-z0)}i1⋯ip={A[0]+A[1]⊛(*)(z-z0)+(A[2]⊛(*)(z-z0))⊛(*)(z-z0)+((A[3]⊛(*)(z-z0))⊛(*)(z-z0))⊛(*)(z-z0)+⋯}i1⋯ip,(2.10)Si1⋯iq(z)={B[0]+∑α≥1P(*)dual[α](B[α],z-z0)}i1⋯iq={B[0]+B[1]⊛(*)dual(z-z0)+(B[2]⊛(*)dual(z-z0))⊛(*)dual(z-z0)+((B[3]⊛(*)dual(z-z0))⊛(*)dual(z-z0))⊛(*)dual(z-z0)+⋯}i1⋯iq,z∈W, A[α] constants of type (p,0), B[α] constants of type (0,q). The set of all in region W*-analytic tensors T and S for a fixed z0 is denoted as 𝔗p(W;*)(z0) and 𝔗q(W;*)(z0).In other words,*-analytic tensor fields are Taylor series in respect of the (dual) *-generalized power function, that is, the Taylor coefficients must take a special place within the “chain structure’’ of the series. If the series above are infinite, we need a criterion for convergence. Of course convergence depends on the region W as well as on the concretely used algebra (ℝn;*). For a simple convergence criterion, we introduce in the space (ℝn)p the following norm. (Since  the vector space ℝn is self-dual, that is, ℝn=(ℝn)*, we can identify (ℝn)p with (ℝn)p=((ℝn)*)p, and the following norm also is a norm for elements Y∈(ℝn)p with ∥Y∥:=:∑i1i2⋯ip|Yi1i2⋯ip|,p≥1)
(2.11)∥X∥:=:∑i1i2⋯ip|Xi1i2⋯ip|,X∈(ℝn)p,p≥1.Proposition 2.5.
Given the norm (2.11) and the denotations of  from above, then the series (2.9) is absolutely convergent in the regionW⊂ℝnif
(2.12)|C(*)kjl|≤Ml,∀j,k∈I1,n,with∑l=1nMl≤1p,∑α≥0∥A[α]∥∥z-z0∥α<∞,z,z0∈W.
The series (2.10) is absolutly convergent in the region W⊂ℝn if
(2.13)|C(*)kjl|≤Nk,∀j,l∈I1,n,with∑k=1nNk≤1q,∑α≥0∥B[α]∥∥z-z0∥α<∞,z,z0∈W.We sum up in the following.Until now two concepts have been introduced: the concept of*-associated Products⊛(*),⊛(*)dualand the concept of*-analytic tensor fields. We have seen  that the operations⊛(*),⊛(*)dualalways have some simple symmetries without restricting the underlying*-operation, and that specializing of*cannot induce the classical symmetries of associativity, commutativity, and Jacobi-symmetry for the operations⊛(*),⊛(*)dual.By means of the*-associated products, the so-called*-analytic tensor fields have been defined. These tensor fields always have symmetries without restricting the underlying*-operation, that is,*-analyticity is a stronger symmetry than smoothness.


## 2.2.*-Analyticity in PAk-Structures. Nonstandard Function Theory

Now we will define such symmetries of an algebra(ℝn;*), so that the corresponding *-analytical tensor fields become interesting “nonstandard symmetries”. In particular, we are interested in symmetries of the partial derivatives, the covariant derivatives, and the Lie-derivatives of *-analytical tensor fields.Definition 2.6.
An algebraic structure(ℝn;*1,*2) is called pseudoassociative of type k (PAk-structure), k = 1,2, if the operations *1,*2 satisfy the following commutation relation:
(2.14)k=1:(x*1y)*2z=(x*1z)*2y,k=2:x*1(y*2z)=z*1(y*2x),forallx,y,z∈ℝn. In the special case *1=*2=*, we denote these structures as pseudorings of typek.It is easy to see that a pseudoring of arbitrary type is associative if and only if it is commutative and a pseudoring with left unit is always commutative.Proposition 2.7.
An algebraic structure(ℝn;*1,*2)is a PAk-structure,  k=1,2, if and only if the structure constant tensors  C(*1) and  C(*2) satisfy the following commutation relation  (The following index symbol “” is used as an index commutator.): (2.15a)k=1:C(*1)ijkC(*2)klm=C(*1)ilkC(*2)kjm,(equivalentwithC(*1)i[jkC(*2)|k|l]m=0),(2.15b)k=2:C(*2)ijkC(*1)lkm=C(*2)ilkC(*1)jkm,(equivalentwithC(*2)i[jkC(*1)l]km=0.)This Proposition allows a characterization of PAk-structures by means of the corresponding structure constant tensors. The problem of the explicit construction of PAk- structure constant tensors will be solved later. Now we will analyze some relationships between the inverse of a matrix (written as()-1) and the (left/right) inverse elements of a “corresponding” vector. We will see that such relationships exist for matrices and vectors derived from PAk-structures.Proposition 2.8.
Given a PA1-structure𝔛=(ℝn;*1,*2) with left unit e(*2,L) and a PA2-structure 𝔜=(ℝn;*1,*2) with right unit e(*1,R), then it holds for arbitrary vectors x∈ℝn:
(2.16)inalgebra𝔛:(x*1)-1=*2ywithx*1y=e(*2,L),inalgebra𝔜:(x*2)-1=y*1withx*2y=e(*1,R).In pseudorings the relations above take the very simple form:(2.17)Pseudoringoffirsttypewithleftunit:(x*)-1=*x-1(*,R,L),Pseudoringofsecondtypewithrightunit:(x*)-1=x-1(*,R,R)*,Commutativepseudoringwithunit:(x*)-1=*x-1(*).In the next step we will show that beside the algebraic characterization there also exists a geometrical characterization of PAk-structures. we will restrict here to a geometrical interpretation only of the pseudoringsymmetries of first type and look at a differentiable manifold𝔐 with an affine connection. As well-known, the corresponding curvature tensor Ω can be written in local coordinates as(2.18)Ωiljm=Γij,lm-Γil,jm+ΓijkΓklm-ΓilkΓkjm=Γi[j,l]m+Γi[jkΓ|k|l]m,
where Γimj denote the affine connection coefficients in local coordinates  (To avoid misunderstandings, we refer to the definitions and denotations given in the appendix: the curvature tensor of a manifold 𝔐 (or a tangent bundle) with a connection is denoted with Ω and its coefficients with Ωiljm, whereas in the more special case of a Riemannian manifold the curvature tensor is denoted as R and its coefficients with Riljm). On the right side, the second summand agrees formal with the left side of the relation (2.15a) for the simple case *1=*2, that is, for pseudorin gsymmetry of first type. This formal similarity is growing in the case of a flat manifold. In this case it holds Ωiljm=0 and if we choose a local coordinate system with the special symmetries (It can be shown, that coordinate systems with this symmetry exist and can be constructed explicitly) Γi[j,l]m=0, then the relation above reduces to Γi[jkΓ|k|l]m=0, which is formal identical with the symmetry (2.15a) of a pseudoring of first type. Therefore we seethat the curvature tensor prefers the symmetry of pseudoassociativity of first type and not the symmetry of associativity. (Associativity follows only in the case of flat spaces with torsion zero, that is, if Γimj=Γmij. In this case, Γimj shows the symmetry of the structure constant tensor of a commutative algebra and according to the considerations above it holds: commutativity + pseudoringsymmetries = commutativity + associativity).Since the pseudoring symmetry of first type seems geometric fundamental, we can expect very interesting symmetries of the corresponding*-analytic tensors. As remarked in the introduction of this section, we are interested especially in the partial derivatives, the covariant derivatives and the Lie derivatives of *-analytical tensor fields. All these derivatives can be written by means of the so called *-derivative, which will be defined now as follows.Definition 2.9.
GivenT∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0). If the following series expansions
(2.19)(𝔇(*)T)i1⋯ip(z):={A[1]+∑α≥2P(*)[α-1](αA[α],z-z0)}i1…ip={A[1]+2A[2]⊛(*)(z-z0)+(3A[3]⊛(*)(z-z0))⊛(*)(z-z0)+⋯}i1…ip,(𝔇(*)S)i1⋯iq(z):={B[1]+∑α≥2P(*)dual[α-1](αB[α],z-z0)}i1⋯iq={B[1]+2B[2]⊛(*)dual(z-z0)+(3B[3]⊛(*)dual(z-z0))⊛(*)dual(z-z0)+⋯}i1⋯iq
are convergent for z∈W, the uniquely defined elements 𝔇(*)T∈𝔗p(W;*)(z0) and 𝔇(*)S∈𝔗q(W;*)(z0) will be denoted as the *-derivative of tensorT and the *-derivative of tensorS.We also will write sometimesT′ instead of 𝔇(*)Tand analog for tensor S. From this definition and relation (2.6c) from Proposition 2.2, it follows immediately that(2.20)Tis(anti)symmetric⇔𝔇(*)Tis(anti)symmetric,Sis(anti)symmetric⇔𝔇(*)Sis(anti)symmetric.
Of course it is not ensured that this formal concept of *-derivative has the properties of the classical derivatives as to be a “direction-independent” limit of a difference quotient, and so forth. The concrete circumstances sensitive depend on the structure of the given algebra. To get interesting symmetries (we have to define what is interesting!) of the *-derivative, we will specialise the algebra to a pseudoring of first type.Theorem 2.10.
Given a pseudoring of first type(ℝn;*) and a tensor  T∈𝔗p(W;*)(z0),W⊆ℝn,p≥1. Then the partial derivatives of  T satisfy the symmetry
(2.21)T,mi1⋯ip=(T′)li2⋯ipC(*)lmi1+(T′)i1li3⋯ipC(*)lmi2+⋯+(T′)i1i2⋯ip-1lC(*)lmip.
If there exists a right unit  e(*,R) on  (ℝn;*) an inverse relation holds:
(2.22)(T′)i1⋯ip=1pT,mi1⋯ipe(*,R)m.Proposition 2.11.
Let(ℂ;*) denote the field of complex numbers. Then the relations (2.21) take the form of the classical Cauchy-Riemannian differential equations.Therefore, our considerations generalize the circumstances of classical complex function theory (and also those of modern function theory built over fields or division algebras) as follows:(2.23)fieldstructureofℂ→pseudoringstructureof(ℝn;*)dimension2ofℝ2→arbitrarydimensionnvectorfieldsonℝ2→tensorfieldsonℝn.
As the most remarkable generalization seems the generalization of the field structure to the structure of a nonassociative noncommutative pseudoring structur with zero divisors, these circumstances legitimize the denotation of (2.21) as Pseudo-Cauchy-Riemannian equations or shorter as PCRE. The analogon of Theorem 2.10  for tensors S∈𝔗q(W;*)(z0) is given as follows.Theorem 2.12.
Given a pseudoring of first type(ℝn;*) and a tensor  S∈𝔗q(W;*)(z0),W⊆ℝn,q≥1, then the partial derivatives of  S satisfy the symmetry (The following equations also will be denoted as PCRE.)
(2.24)Si1⋯iq,m=(S′)li2⋯iqC(*)i1ml+(S′)i1li3⋯iqC(*)i2ml+⋯+(S)i1i2⋯iq-1lC(*)iqml.
If there exists a right unit  e(*,R) of (ℝn;*), an inverse relation holds:
(2.25)(S′)i1⋯iq=1qSi1⋯iq,me(*,R)m.Now we will study the covariant derivative of*-analytical tensor fields. For this it is necessary to formalize the PCRE. By considering the properties of the *-associated products ⊛(*) and ⊛(*)dual from Proposition 2.2, we see the possibility of the following formulation:(2.26)∂T=𝔒(*)T⊛(*),∂S=𝔒(*)S⊛(*)dual.
This relations show that partial derivatives can be “transformed” onto “ordinary” *-derivatives. This will offer an interesting approach for solving partial differential equations by transforming these equations onto ordinary “*-differential equations”, that is, ordinary algebra-differential equations.For studying the covariant derivatives of*-analytical tensor fields we remember that for arbitrary tensors T∈𝔗p(W),S∈𝔗q(W),W⊂ℝn the covariant derivatives in component form are defined by(2.27)T;ji1i2⋯ip=T,ji1i2⋯ip+Tki2⋯ipΓkji1+Ti1ki3⋯ipΓkji2+⋯+Ti1i2⋯ip-1kΓkjip,Si1⋯iq;j=Si1⋯iq,j-Ski2⋯iqΓi1jk-Si1ki3⋯iqΓi2jk-⋯-Si1i2⋯iq-1kΓiqjk,
where Γjki denotes the Christoffel symbols of the chosen ℝn-coordinate system. Comparison with the relations of Proposition 2.2 shows a strong formal similarity which legitimates the following denotations:(2.28)(T⊛(Γ))i1i2⋯ip:=Tki2⋯ipΓkji1+Ti1ki3⋯ipΓkji2+⋯+Ti1i2⋯ip-1kΓkjip,(S⊛(Γ)dual)i1⋯iq:=Ski2⋯iqΓi1jk+Si1ki3⋯iqΓi2jk+⋯+Si1i2⋯iq-1kΓiqjk.Proposition 2.13.
Given a pseudoring of first type(ℝn;*)and tensors T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,p,q≥1, then the covariant derivatives  DT,DS satisfy the following symmetries:
(2.29)DT=𝔇(*)T⊛(*)+T⊛(Γ),DS=𝔇(*)S⊛(*)dual-S⊛(Γ)dual.Now we will study the Lie-derivatives of*-analytical tensor fields. For this we remember on the Lie-derivative 𝔏v:𝔗qp(W)→𝔗qp(W) of a tensor T∈𝔗qp(W) in respect to a vector field v:(2.30)(𝔏vT)j1⋯jqi1⋯ip=Tj1⋯jq,mi1⋯ipvm-Tj1⋯jqmi2⋯ipv,mi1-Tj1⋯jqi1mi3⋯ipv,mi2-⋯-Tj1⋯jqi1⋯ip-1mv,mip+Tmj2⋯jqi1⋯ipv,j1m+Tj1mj3⋯jqi1⋯ipv,j2m+⋯+Tj1⋯jq-1mi1⋯ipv,jqm.Proposition 2.14.
Given two pseudorings of first type(ℝn;*) and  (ℝn;∘), furthermore the tensors T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,p,q≥1and a vector v∈𝔗1(W;∘)(z0). Then the Lie-derivatives 𝔏vTand  𝔏vSsatisfy the following symmetries (The following operation  ∘T is defined by  C(∘T)mjk=C(∘)jmk, that is, by “transposition” of the corresponding quadratic matrix with fixed index  k.):
(2.31)𝔏vT=𝔇(*)T⊛(*)v-T⊛(∘T)𝔇(∘)v,𝔏vS=𝔇(*)S⊛(*)dualv+S⊛(∘T)dual𝔇(∘)v.
For a pseudoring of first type  (ℝn;*) with right unit  e(*,R) and a tensor field, it holds that (2.32a)𝔇(*)T=𝔏e(*,R)T,T∈𝔗p(W;*)(z0),(2.32b)(𝔏gf)′=𝔏gf′+𝔏g′f,f,v∈𝔗1(W;*)(z0).Relation (2.32a) shows a very simple characterization of the *-derivative by the Lie-derivative, but it is only possible in a pseudoring with a right unit.After the partial derivatives, covariant derivatives and Lie-derivatives we will study the exterior derivatives of*-analytic forms. Let us denote the set of all in region W⊆ℝn smooth q -forms with 𝔄q(W), and the subset of all in W*-analytical q-forms with 𝔄q(W;*)(z0).Proposition 2.15.
Given a pseudoring(ℝn;*) of first type, an arbitrary nontrivial q-form ω∈𝔄q(W;*)(z0),and arbitrary vector fields u,v∈𝔗1(W;*)(z0),w∈𝔗1(W;*)(z0),v not constant (“Not constant” means of course not constant in the considered coordinate system), then it holds that
(2.33)d⋀ω=0⇔(ℝn;*)iscommutative,u*v∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative,w*dualv∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative,u-1(*)∈𝔗1(W;*)(z0)⇔(ℝn;*)iscommutative.This proposition shows that some fundamental symmetries with practical importance only exist in commutative pseudorings: the*-product of *-analytical vector fields is *-analytic only for commutative pseudorings, and an analogue result holds for the *dual-product. Also the inverse element of a *-analytical vector field will be *-analytical only in a commutative pseudoring. Let us sum up the relations for the classical derivatives as follows:Pseudo-Cauchy-Riemann equations:(2.34a)∂T=𝔇(*)T⊛(*),∂S=𝔇(*)S⊛(*)dual,
Covariant derivatives:(2.34b)DT=𝔇(*)T⊛(*)+T⊛(Γ),DS=𝔇(*)S⊛(*)dual-S⊛(Γ)dual,
Lie-derivative:(2.34c)𝔏vT=𝔇(*)T⊛(*)v-T⊛(∘T)𝔇(∘)v,𝔏vS=𝔇(*)S⊛(*)dualv+S⊛(∘T)dual𝔇(∘)v,𝔇(*)T=𝔏e(*,R)T,
Exterior  derivative:
(2.34d)d⋀ω=0⇔(ℝn;*)iscommutative,ω∈𝔄q(W;*)(z0),with (ℝn;*),(ℝn;∘) pseudorings of first type, T∈𝔗p(W;*)(z0),S∈𝔗q(W;*)(z0),W⊂ℝn,v∈𝔗1(W;∘)(z0). We sum up in the following paragraph.Our aim was to study*-analytical tensor fields in the framework of pseudorings, which have been introduced here as a fundamental concept. It was shown that pseudoringsymmetry is an algebraic symmetry which also appears in the world of differential geometry, moreover: the geometrical world prefers the pseudoringsymmetry against the classical symmetry of associativity. For this the study of pseudorings is legitimized, and so the study of tensor fields built over pseudorings. In particular we have studied the partial derivatives of*-analytical tensor fields, their covariant derivatives, their Lie-derivatives and their exterior derivatives, all this in the frame of pseudoringsymmetry. It was shown, that all these fundamental derivatives can be written in terms of the so-called*-derivative, which has been introduced here as a fundamental concept, following from the symmetry of*-analyticity.


## 2.3. Generalizations. The World of GAPs

Until now we have dealt with PAk-structures(ℝn;*1,*2) and pseudorings (ℝn;*), that is, with algebraic structures defined by maximal two product operations. For some later applications, this framework will be too small and shall be generalized here (The most applications on mathematical physics only need some small parts of this section: the concept of a GAP, of a GAP characteristic, and of GAP Exponentials). Such an algebra generalization will lead us to the concept of GAPs (PAk-structures and pseudorings will be shown as the most simple GAP-structures) and will allow us to generalize the function theoretical concepts of *-analyticity, *-derivative, *-integration, and so forth, from the section above by replacing the underlying PAk-structures (pseudorings) by GAPs.The aim of this section is to generalize the concepts of PAk-structures,*-analyticity,*-derivative,*-integration, Pseudo Cauchy Riemann equations, and so forth.Let us denote𝔛=(ℝn;*1,*2,…,*M) as an algebraic structure with the operations *α:ℝn×ℝn→ℝn,α∈I1,M,M an arbitrary natural number which also might be infinite. To generalize the concept of *-analyticity we have to generalize the concept of the power function from Section 2.1. For this we introduce the maps P(*1,*2,…,*α)[α]:(ℝn)p×(ℝn)→(ℝn)p and P(*1,*2,…,*α)dual[α]:(ℝn)q×(ℝn)→(ℝn)q as follows:(2.35)P(*1,*2,…,*α)[α](A,z):=(⋯((A⊛(*α)z)⊛(*α-1)z)⋯⊛(*2)z)⊛(*1)z,α-times,P(*1,*2,…,*α)dual[α](B,z):=(⋯((B⊛(*α)dualz)⊛(*α-1)dualz)⋯⊛(*2)dualz)⊛(*1)dualz,α-times,A∈(ℝn)p,B∈(ℝn)q arbitrary constants, z∈ℝn.  After these assumption we can introduce the concept of “chain-analyticity”.Definition 2.16.
Given an arbitrary algebraic structure𝔛=(ℝn;*1,*2,…,*M) and arbitrary tensors T∈𝔗p(W1),S∈𝔗q(W1), we denote T and S, respectively, as chain-analyticin region W1, if there exists an element z0∈W1, which allows the following representations of T, respectively, S:
(2.36)Ti1⋯ip(z)={A[0]+∑α=1MP(*1,*2,…,*α)[α](A[α],z-z0)}i1⋯ip:=A[0]+A[1]⊛(*1)(z-z0)+(A[2]⊛(*2)(z-z0))⊛(*1)(z-z0)+((A[3]⊛(*3)(z-z0))⊛(*2)(z-z0))⊛(*1)(z-z0)+⋯,Si1⋯iq(z)={B[0]+∑α=1MP(*1,*2,…,*α)dual[α](B[α],z-z0)}i1⋯iq:=B[0]+B[1]⊛(*1)dual(z-z0)+(B[2]⊛(*2)dual(z-z0))⊛(*1)dual(z-z0)+((B[3]⊛(*3)dual(z-z0))⊛(*2)dual(z-z0))⊛(*1)dual(z-z0)+⋯,z∈W1,A[α] a constant of type (p,0),B[α] a constant of type (0,q). The set of all in region W1 chain-analytical tensors T and S for fixed z0 is denoted as 𝔗p(W1;*1,*2,…,*M)(z0) and 𝔗q(W1;*1,*2,…,*M)(z0).Consider that in the series above an infinite value ofM is allowed and an infinite set of *α-operations is allowed to contain an infinite subset of identical operations. In the case *1=*2=⋯=*M  (=finite or infinite), we will denote the structures above as *-analytical as in the earlier case of a single product operation. The denotation “chain-analytical” comes from the fact, that the series above are built by *α-operations like the “elements of a chain”. We see, that for different *α-operations the structure of chain-analytical tensor fields is much more general than the structure of simple *-analytical tensor fields. Now we will generalize the concept of PAk-structures for vector spaces with different dimensions.Now I will present a concept, which allows a short overview about all symmetries—PAk-symmetry as well as standard symmetries—of a given algebraic structure. For this we assign the algebraic structure𝔛=(ℝn;*1,*2,…,*M) a M×M-matrix G(𝔛), denoted as the symmetry characteristic of the algebra𝔛 as follows:(2.37)G(𝔛)αβ:={kifthesubalgebra(ℝn;*α,*β)isaPAk-structure,0otherwise.}
(“0” means: “no PAk symmetry is assumed” and not: “no PAk-symmetry is allowed”. With other words: In the case “0” it is possible that a PAk-symmetry exists, but it is not ensured.)Definition 2.17.
An algebraic structure𝔛=(ℝn;*1,*2,…,*M)is called a GAP of order M, if the symmetry characteristic G(𝔛) includes at least one value 1 or 2. In this case G(𝔛) is denoted as the GAP characteristic of the GAP 𝔛.In other words a GAP is an algebraic structure which has at least one PA1-symmetry or one PA2 -symmetry. For further applications it will be advantageable, to generalize the GAP-characteristic by giving an overview not only about the PAk-symmetries but also about the ”standard symmetries” of all algebraic structures(ℝn;*α), that is, of its possible (anti)commutativity, Jacobi-symmetry, associativity, existence of a right unit e(*,R), existence of a left unit e(*,L), and so forth. Since these symmetries all are defined for a single algebra operation, they only will appear in the main diagonal of G(𝔛). Concretely we introduce the following denotations as subscripts of the elements G(𝔛)αα:(2.38)c~commutativity,ac~anticommutativity,J~Jacobi-symmetry,e(R)~existenceofarightunit,e(L)~existenceofaleftunit,e~existenceofaunit.
As an example, we consider an algebraic structure 𝔛=(ℝn;*1,*2,*3,*4) with the following symmetry characteristic:(2.39)G(𝔛)=(010000e(R)10000ac,J00200)
which means that the algebra (ℝn;*2) has a right unit, the algebra (ℝn;*3) is anticommutative and satisfies the Jacobi-symmetry (i.e., (ℝn;*3) is a Lie-algebra), the algebra (ℝn;*4) is commutative, the algebraic structure (ℝn;*1,*2) is a PA1-structure, and the algebraic structure (ℝn;*4,*2) is a PA2-structure. Per definition 𝔛 is a GAP.Now some special types of GAPs will be introduced which allow a wide generalization of the functional theoretical concepts of*-analyticity, *-derivative, Pseudo Cauchy Riemann equations, *-integration, and so forth.Definition 2.18.
A PA1-chain is a GAP𝔛=(ℝn;*1,*2,…,*M) with PAkαα-1=1,α∈I2,M. A PA1-chain is denoted as a closed PA1-chain, if additionally to the above it holds  PAk1M=1. A closed PA1-chain of Lie-typeis defined as a closed PA1-chain of order 3 with the additional restriction
(2.40)(x*3y)*2z+(y*2z)*1x+(z*1x)*3y=0,∀x,y,z∈ℝn.The concept of a PA1-chain allows a wide generalization of the earlier concept of the*-derivation, which will be presented now.Definition 2.19.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and chain-analytic tensors T∈𝔗p(W1;*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0), then the following series expansions
(2.41)(𝔇(*1)T)i1⋯ip(z):={A[1]+∑α≥2MP(*2,…,*α)[α-1](αA[α],z-z0)}i1⋯ip:={A[1]+2A[2]⊛(*2)(z-z0)+(3A[3]⊛(*3)(z-z0))⊛(*2)(z-z0)+⋯}i1⋯ip,(𝔇(*1)S)i1⋯iq(z):={B[1]+∑α≥2MP(*2,…,*α)dual[α-1](αB[α],z-z0)}i1⋯iq:={B[1]+2B[2]⊛(*2)dual(z-z0)+(3B[3]⊛(*3)dual(z-z0))⊛(*2)dual(z-z0)+⋯}i1⋯iq,z∈W1, uniquely define elements 𝔇(*1)T∈𝔗p(W1;*2,…,*M)(z0) and 𝔇(*1)S∈𝔗q(W1;*2,…,*M)(z0) which will be denoted as the *1-derivative of tensorT and the *1-derivative of tensorS.We see that the tensors𝔇(*1)T and 𝔇(*1)S do not contain the operation *1 any longer and so we cannot define 𝔇(*1)𝔇(*1)T or 𝔇(*1)𝔇(*1)S. Only 𝔇(*2)𝔇(*1)T and 𝔇(*2)𝔇(*1)S make sense, and also 𝔇(*3)𝔇(*2)𝔇(*1)T and 𝔇(*3)𝔇(*2)𝔇(*1)S, and so forth. Thus we will define the operators(2.42)𝔇(*k,*k-1,…,*1)k:=𝔇(*k)𝔇(*k-1)⋯𝔇(*1),k∈I1,M
as the chain derivatives of order k. In the special case *1=*2=⋯=*k:=* we write simplifying 𝔇(*)k instead of 𝔇(*,*,…,*)k. From the  definitions above  it follows for z∈W1:(2.43)(𝔇(*k,*k-1,…,*1)kT)i1⋯ip(z):={A[k]+∑α≥k+1MP(*2,…,*α)[α-k](α(α-1)⋯(α-k)A[α],z-z0)}i1⋯ip,(𝔇(*k,*k-1,…,*1)kS)i1⋯iq(z):={B[k]+∑α≥k+1MP(*2,…,*α)dual[α-k](α(α-1)⋯(α-k)B[α],z-z0)}i1⋯iq,
and we see that(2.44)𝔇(*k,*k-1,…,*1)kT∈𝔗p(W1;*k+1,…,*M)(z0),𝔇(*k,*k-1,…,*1)kS∈𝔗q(W1;*k+1,…,*M)(z0).Theorem 2.20.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and a chain-analytic tensorT∈𝔗p(W1;*1,…,*M)(z0),p≥1. Then the following symmetries hold:
(2.45)(𝔇(*k,*k-1,…,*1)kT),αi1⋯ip=(𝔇(*k+1,*k,…,*1)k+1T)li2⋯ipC(*k+1)lαi1+(𝔇(*k+1,*k,…,*1)k+1T)i1li3⋯ipC(*k+1)lαi2+⋯+(𝔇(*k+1,*k,…,*1)k+1T)i1i2⋯ip-1lC(*k+1)lαip,forallα∈I1,n1,i1,…,ip∈I1,n2,k∈I0,M-1 (In the case k=0 the left side must be interpreted as T,αi1⋯ip). An inverse relation exists, if the algebra (ℝn;*1) has a right unit  e(*1,R):
(2.46)(D(*k+1,*k,…,*1)k+1T)i1i2i3⋯ip=1p(𝔇(*k,*k-1,…,*1)kT),mi1⋯ipe(*1,R)m.Theorem 2.21.
Given a PA1-chain𝔛=(ℝn;*1,*2,…,*M) and a chain-analytic tensor S∈𝔗q(W1;*1,…,*M)(z0),p≥1. Then the following symmetries hold:
(2.47)(𝔇(*k,*k-1,…,*1)kS)i1⋯iq,α=(𝔇(*k+1,*k,…,*1)k+1S)li2⋯iqC(*k+1)i1αl+(𝔇(*k+1,*k,…,*1)k+1S)i1li3⋯iqC(*k+1)i2αl+⋯+(𝔇(*k+1,*k,…,*1)k+1S)i1i2⋯iq-1lC(*k+1)iqαl,forallα∈I1,n1,i1,…,ip∈I1,n2,k∈I1,M-1 (In the case k=0 the left side must be interpreted as Si1⋯iq,α). An inverse relation exists, if the algebra (ℝn;*1) has a right unit  e(*1,R):
(2.48)(𝔇(*k+1,*k,…,*1)k+1S)i1i2⋯iq=1q(𝔇(*k,*k-1,…,*1)kS)i1⋯iq,me(*1,R)m.The proofs of Theorems2.20 and 2.21 run the same lines as the proofs of Theorems 2.10 and 2.12. Relations (2.45) and (2.47) will be denoted as Pseudo Cauchy Riemann equations (PCRE) as in the simple case of a single vector space and a single product operation. (The expression𝔇(*0,*0-1,…,*1)0T must be interpreted as T, and 𝔇(*0,*0-1,…,*1)0S as S.) Now we will generalize the former concept of *-integration.Definition 2.22.
Given an arbitrary algebra𝔛=(ℝn;*1,*2,…,*M)and chain-analytic tensorsT∈𝔗p(W1;*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0),p≥1, and further let (ℝn;*0) an algebra with the property
(2.49)(ℝn;*1,*0)isaPA1-structure,
Then the series
(2.50)(𝔇(*0)-1T)i1⋯ip(z):={∑α≥0MP(*0,*1,…,*α)[α+1](1α+1A[α],z-z0)}i1⋯ip+Ki1⋯ip:={K+A[0]⊛(*0)(z-z0)+(12A[1]⊛(*1)(z-z0))⊛(*0)(z-z0)+((13A[2]⊛(*2)(z-z0))⊛(*1)(z-z0))⊛(*0)(z-z0)+⋯}i1⋯ip,(𝔇(*0)-1S)i1⋯iq(z):={∑α≥0MP(*0,*1,…,*α)dual[α+1](1α+1B[α],z-z0)}i1⋯iq+Li1…iq:={L+B[0]⊛(*0)dual(z-z0)+(12B[1]⊛(*1)dual(z-z0))⊛(*0)dual(z-z0)+((13B[2]⊛(*2)dual(z-z0))⊛(*1)dual(z-z0))⊛(*0)dual(z-z0)+⋯}i1⋯ip,
with z∈W1,K and L arbitrary constants, are called the *0-integrals or the *0-antiderivatives of the tensors T and S.The denotation𝔇(*0)-1 is well chosen because it is 𝔇(*0)𝔇(*0)-1T=𝔇(*0)-1𝔇(*0)T and 𝔇(*0)𝔇(*0)-1S=𝔇(*0)-1𝔇(*0)S. We see, that(2.51)T∈𝔗p(W1;*1,…,*M)(z0)⇒𝔇(*0)-1T∈𝔗p(W1;*0,*1,…,*M)(z0),S∈𝔗q(W1;*1,…,*M)(z0)⇒𝔇(*0)-1S∈𝔗q(W1;*0,*1,…,*M)(z0).Proposition 2.23.
If𝔛is a PA1-chain, then the  *0-integrals  𝔇(*0)-1Tand 𝔇(*0)-1S can be expressed by the classical integral conception as follows:
(2.52)𝔇(*0)-1T(z)=∫T(z)⊛(*0)dz,𝔇(*0)-1S(z)=∫S(z)⊛(*0)dualdz.Theorem 2.24.
𝔛a PA1-chain and  𝔏 a smooth closed curve in region  W1. Then it is
(2.53)∮𝔏T(z)⊛(*0)dz=0,∮𝔏S(z)⊛(*0)dualdz=0.This Theorem generalizes the well-known classical Cauchy Integral Theorem for GAPs and is denoted asPseudo Cauchy Integral Theorem (PCIT). For the next step we remember on the classical denotation y′=dy/dx for a scalar differentiable function y(x), or formally y′dx=dy. It shall be shown now, that for the *-derivative of vector-valued functions a similar result holds:Proposition 2.25.
Given a PA1-chain𝔛=(ℝn;*0,*1,…,*M)and vectorsU∈𝔗1(W;*0,*1,…,*M)(z0),V∈𝔗1(W;*0,*1,…,*M)(z0). Then it holds
(2.54)dU(z)=𝔇(*0)U⊛(*0)dz,dV(z)=𝔇(*0)V⊛(*0)dualdz.Proof.
We start from the relation (2.52), with T∈𝔗1(W1;*1,…,*M)(z0) and S∈𝔗1(W1;*1,…,*M)(z0). Now we define U(z)=𝔇(*0)-1T(z),V(z)=𝔇(*0)-1S(z), from which follows U∈𝔗1(W;*0,*1,…,*M)(z0),V∈𝔗1(W;*0,*1,…,*M)(z0) as it is assumed above. Inserting U and V in (2.52) we obtain U(z)=∫𝔇(*0)U⊛(*0)dz,⇔dU(z)=𝔇(*0)U⊛(*0)dz, and analog:V(z)=∫𝔇(*0)V⊛(*0)dualdz,⇔dV(z)=𝔇(*0)V⊛(*0)dualdz.We sum up in the following paragraphIn this section, the main earlier results have been generalized,along with the algebraic structures as well as the tensor fields built over these structures. In particular the PAk-structures have been generalized by the concept of the so-called “GAPs”. We have investigated special GAPs like PA1 chains and their specializations (closed PA1-chains and closed PA1-chains of Lie-type), and have shown that the main results of the classical complex function theory (Cauchy Riemannian differential equations, Cauchy Integral Theorem) also hold in PA1-chains.


## 2.4. The Explicit Construction of GAPs

The aim of this section is the explicit construction of a wide variety of GAPs.In other words, we have to construct structure constant tensors of pseudorings, PAk-structures, and PA1-chains explicitly. This problem makes sense, because without knowledge of concrete GAPs, the results of the sections above would be not applicable in practice. In the following, we introduce the mapsHn:ℤ→]0,n], n∈ℕ (ℤ the set of integers, ℕ the set of natural numbers) with n≥1 by(2.55)Hn(x)={x-mn,ifmn<x≤(m+1)n,xif0<x≤n,x+mnif-mn<x≤(-m+1)n,
where m∈ℕ denotes the smallest possible value, defined uniquely by the variable x and the conditions above (For example, let us consider the case n=3,x=3.7. Then m is uniquely defined by m=1, since it is mn<x≤(m+1)n only for m=1. Then it is Hn(x)=x-mn=0.7). From this definition it follows immediately Hn(x+y)=Hn(y+x), and  Hn(x+Hn(y+z))=Hn(y+Hn(z+x))=Hn(z+Hn(x+y))=Hn(x+y+z).Proposition 2.26.
Given the constantsP∈ℕ,M∈I1,P,andAM∈ℝ,aM,σ∈ℤ.Then PAk-structures (ℝn;*1,*2) are given by the following structure constant tensors:(2.56a)casek=1:C(*1)ijk=∑M=1PAMδHn(aMi+σj)k,C(*2)klm=δHn(k+σl)m,(2.56b)casek=2:C(*1)ijk=δHn(σi+j)k,C(*2)klm=∑M=1PAMδHn(aMk+σl)m.Proof.
We show that the algebra (2.56a) satisfies the PA1-condition C(*1)i[jkC(*2)|k|l]m=0. Writing all sums explicitely we get ∑k=1nC(*1)ijkC(*2)klm=∑k=1n∑M=1PAMδHn(aMi+σj)kδHn(k+σl)m=∑M=1PAMδHn(Hn(aMi+σj)+σl)m, because δHn(aMi+j)k only delivers a nontrivial value for the index value k=Hn(aMi+j) and this index k must be inserted in the term δHn(k+σl)m. Since it is Hn(Hn(aMi+σj)+σl)=Hn(aMi+σj+σl), the right side above is symmetrical in indices j and l. Thus the left side is symmetrical in indices j and l, that is, the PA1-symmetry holds and statement (2.56a) has been proved. Now we have to show that algebra (2.56b) satisfies the PA2-condition C(*2)i[jkC(*1)l]km =0:∑k=1nC(*2)ijkC(*1)lkm =∑k=1n∑M=1PAMδHn(aMi+σj)kδHn(σl+k)m = ∑M=1PAMδHn(σl+Hn(aMi+σj))m = δHn(σl+aMi+σj)m. The right side is symmetrical in indices j and l, and so the left side is, that is, the PA2-symmetry holds and statement (2.56b) has been proved.Proposition 2.27.
Let ben∈ℕ,σ∈ℤarbitrary. Then the structure constant tensor
(2.57)C(*)ijk=δHn(i+σj)k,i,j,k∈I1,n
defines a pseudoring of first type  (ℝn;*). In the case  σ=1, a unit element exists, given by  e(*)j=δnj, and furthermore it hold s that(2.58a)∃λ∈ℝn,λ≠0with∄λ-1(*),(2.58b)Anonexistingλ-1(*)doesnotimplyλ2=0.Because the PAk-structures above are defined by Kronecker-symbols, we will denote them asPAk-structures of Kronecker-type or shorter asKronecker PAk-structures. According to the property (2.58a) a commutative Kronecker pseudoring is not a division algebra. It is easy to show, that this property also holds for noncommutative Kronecker pseudorings. Also the property (2.58a) can be generalized for noncommutative Kronecker pseudorings, where λ-1(*) must be replaced by λ-1(*,R,R), and so forth. (see the definition of λ-1(*,R,R),λ-1(*,R,L),λ-1(*,L,R),λ-1(*,L,L) in (2.2)). We see that a Kronecker algebra has a very simple structure, because the values of the structure constants are given by only two reals: 0 and 1. Now another type of PAk-structures will be presented which is more subtle than the Kronecker type.Proposition 2.28.
Given an algebra(ℝn;*1,*2) as follows:
(2.59)C(*α)ijh=diδjh+b(*α)djδih+didjh(*α)h,α=1,2,(informaldenotation:C(*α)=d⊗I+b(*α)I⊗d+d⊗d⊗h(*α))
with d,h(*1),h(*2)∈ℝn, d≠0,b(*1),b(*2)∈ℝ arbitrary nonzero constants. Then (ℝn;*1,*2) is a PAα-structure if and only if
(2.60)〈d,h(*α)〉=-1+σ(α){b(*2)-b(*1)σ(α)}withσ(α)=(-1)α-1,α=1,2.(ℝn;*α),α=1,2has a right unit  e(*α,R) if and only if
(2.61)〈d,h(*α)〉=-1,Thenitise(*α,R)=-b(*α)-1h(*α).The PAk-structures presented above are built by combinations of rank 2-tensors and rank 1-tensors. Therefore we will denote these structures asPAk-structures of splitting type or shorter as splitting PAk-structures.From this follows immediately the structure of splitting pseudorings. It is easy to show, that Kronecker PAk-structures generally cannot be transformed onto splitting PAk-structures by basistransformations in ℝn, that is, these both structures are really different.Splitting pseudorings have a large advantage to Kronecker pseudorings because it allow the explicit calculation ofzP,z-1, and so forth.Proposition 2.29.
Given the structure constant tensor(2.62)C(*)ijk=diδjk+bdjδik+didjhkwithhkdk=-1,(informaldenotation:C(*α)=d⊗I+bI⊗d+d⊗d⊗h,with〈h,d〉=-1)i,j,k∈I1,n,b∈ℝ only restricted by  b≠0.  Then the following statements hold:Statement 1.
The algebra(ℝn;*)defines ann-dimensional pseudoring of first type with right unite(*,R)=-b-1h.Statement 2.
In this algebra a vectorzhas a right invers elementz-1(*,R,R)in respect of the right unit,if and only if〈z,d〉≠0.Statement 3.
In(ℝn;*)the power functions, roots and inverses of vectors are calculable explicitly as follows:(2.63)z[P]=bP-2{(b+P-1)〈d,z〉P-1z+(P-1)〈d,z〉Ph},P∈ℕ,(2.64)z-1(*,R,R)=-b-2{b〈d,z〉-2z+(1+b)〈d,z〉-1h},
where z[P] is defined by z[P]=(⋯((z*z)*z)⋯)*z. In the case b=1 the relation (2.63) also holds for arbitrary rational numbers.Statement 4.
Forb=1it holds(2.65)Anonexistingλ-1(*)impliesλ2=0,U∈𝔗1(W;*)(z0)⇒λ<d,U>∈𝔗1(W;*)(z0)iff∄λ-1(*),U∈𝔗1(W;*)(z0)⇒d<λ,U>∈𝔗1(W;*)(z0)iff∄λ-1(*).Proposition2.29 shows, that in the framework of splitting pseudorings the inverse elements as well as the roots of vectors are calculable explicitly, that is, without using numerical methods. In Kronecker algebras this is not possible. Now we will show, that PA1-chains and furthermore PA1-chains of Lie-type can be constructed explicitly, proving, that the world of the PAk-structures is compatible with the world of generalized Lie-symmetries:Proposition 2.30.
The following algebraic structures define PA1-chains𝔛=(ℝn;*1,*2,…,*M):Kronecker PA1-chains:(2.66a)C(*α)ijk=δHn(i+σj)k,α=1,…,M,σ∈ℤarbitrary,
Splitting PA1-chains:
(2.66b)C(*α)ijk=diδjk+b(*α)djδik+didjh(*α)k,α∈I1,M,with1+b(*α)+〈d,h(*α)〉=b(*α-1),α∈I2,M.
Modified Splitting
(2.66c)C(*α)ijk=d(*α)iA(*α)jk+gjB(*α)ik,α∈I1,M,withtherestrictions
PA1-chains:
(2.66d)A(*α+1)jkd(*α)k=0,B(*α+1)jkd(*α)k=ud(*α+1)j,A(*α+1)jkB(*α)km=uA(*α)jm,u∈ℝarbitrary.Proposition 2.31.
The following algebraic structure(ℝn;*1,*2,*3) defines a closed PA1-chain from Lie type:
(2.67)C(*α)ijk=diδjk+b(*α)djδik+didjh(*α)k,α∈I1,3,withb(*1)=13(〈h(*2),d〉-〈h(*1),d〉),b(*2)=-1+23〈h(*2),d〉+13〈h(*1),d〉,b(*3)=1+〈h(*1),d〉,h(*1)mb(*2)+h(*2)mb(*3)+h(*3)mb(*1)=0.Until now we have delt with special PAk -structures (Kronecker structures and splitting structures). Now we will analyze the problem, if PAk-structures can be combined in a way, that we come to PAk-structures again. There are two ways to analyze this problem. The first way is to try a construction of newn-dimensional PAk-structure by combining n-dimensional PAk-structures, the second way is to try a construction of new N-dimensional PAk-structure by combining n-dimensional PAk-structures, N>n. We will give answers to both problems by the following both propositions.Proposition 2.32.
Given a GAP𝔛=(ℝn;*1,*2,*3,*4)with the characteristic
(2.68)G(𝔛)=(0010000100000000).
Then the algebra 𝔜=(ℝn;⊛1,⊛2)defined by
(2.69)C(⊛1)ijk:=C(*1)ijk+C(*2)ijk,C(⊛2)ijk:=C(*3)ijk+C(*4)ijk
is a PA1-structure, if and only if
(2.70)C(*2)i[jkC(*3)|k|l]m=-C(*1)i[jkC(*4)|k|l]m.
Given a GAP 𝔛=(ℝn;*1,*2,*3,*4)with the characteristic:
(2.71)G(𝔛)=(0100001000000000).
Then the algebra ℨ=(ℝn;⊚1,⊚2)defined by
(2.72)C(⊚1)ijk:=C(*1)ijsC(*2)spkup,C(⊚2)ijk:=C(*3)ijsC(*4)spkvp,u,v∈ℝnarbitrary
is a PA1-structure.Proof.
To prove the first statement we calculate(2.73)C(⊛1)ijkC(⊛2)klm=(C(*1)ijk+C(*2)ijk)(C(*3)klm+C(*4)klm)=C(*1)ijkC(*3)klm+C(*1)ijkC(*4)klm+C(*2)ijkC(*3)klm+C(*2)ijkC(*4)klm,⇒C(⊛1)i[jkC(⊛2)|k|l]m=C(*1)i[jkC(*3)|k|l]m+C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m+C(*2)i[jkC(*4)|k|l]m=GAP-symmetries(2.69)C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m.
We see, that the PA1 -symmetry C(⊛1)i[jkC(⊛2)|k|l]m=0 holds if and only if C(*1)i[jkC(*4)|k|l]m+C(*2)i[jkC(*3)|k|l]m=0, according to (2.70). To prove the second statement we calculate
(2.74)C(⊚1)ijkC(⊚2)klm=C(*1)ijsC(*2)spkupC(*3)klrC(*4)rqmvq,⇒C(⊚1)i[jkC(⊚2)|k|l]m=C(*1)i[jsC(*2)|spkupC(*3)k|l]rC(*4)rqmvq=GAP-symmetries(2.72)C(*1)i[jsC(*2)|s|l]kupC(*3)kprC(*4)rqmvq=GAP-symmetries(2.72)0.Proposition 2.33.
Given a PA1-structure𝔛=(ℝn;*1,*2). Then the following algebra 𝔜=(ℝ2n;⊛) is a pseudoring of first type:
(2.75)C(⊛)ijk={C(*1)ijk,ifi,j,k∈I1,n,C(*2)i-nj-nk-n,ifi,j,k∈In+1,2n,0,inallothercases.Proof.
Per defnition it is(2.76)C(⊛)ijkC(⊛)klm=∑k=1nC(⊛)ijkC(⊛)klm+∑k=n+12nC(⊛)ijkC(⊛)klm
for arbitrary indices i,j,l,m∈I1,2n. Let us consider first the case that i∈I1,n. Then nontrivial elements of C(⊛)ijk only exist in the case j,k∈I1,n, defined by C(⊛)ijk=C(*1)ijk, and the term above takes the form
(2.77)C(⊛)ijkC(⊛)klm=∑k=1nC(⊛)ijkC(⊛)klm=∑k=1nC(*1)ijkC(⊛)klm={0,ifl>norm>n,∑k=1nC(*1)ijkC(*2)klm,ifl,m∈I1,n.
Thus it follows for i∈I1,n:(2.78)C(⊛)i[jkC(⊛)|k|l]m={0,ifl>norm>n∑k=1nC(*1)i[jkC(*2)|k|l]m,ifl,m∈I1,n}=0
in every case, because the algebra (ℝn;*1,*2) is assumed as a PA1-structure, that is, it is ∑k=1nC(*1)i[jkC(*2)|k|l]m=0 for i∈I1,n.
Let us consider now the case thati∈In+1,2n. Then nontrivial elements of C(⊛)ijk only exist in the case j,k∈In+1,2n, defined by C(⊛)ijk=C(*2)i-nj-nk-n, and we get
(2.79)C(⊛)ijkC(⊛)klm=∑k=n+12nC(⊛)ijkC(⊛)klm=∑k=n+12nC(*1)i-nj-nk-nC(⊛)klm={0,ifl≤norm≤n,∑k=n+12nC(*1)i-nj-nk-nC(*2)k-nl-nm-n,ifl,m∈In+1,2n.
Thus it follows with the denotations ι̅=i-n,j̅=j-n,k̅=k-n,l̅=l-n,m̅=m-n:(2.80)C(⊛)i[jkC(⊛)|k|l]m={0ifl≤norm≤n,∑k̅=1nC(*1)ι̅[j̅k̅C(*2)|k̅|l̅]m̅inallothercases,}=0.
in every case, because the indices ι̅,j̅,k̅,l̅,m̅ move in the region I1,n, and the algebra (ℝn;*1,*2) is assumed as a PA1-structure.In this section a wide variety of GAPs has been constructed explicitly. In particular two different types of explicit calculable PAk-structures have been characterized: Kronecker PAk-structures and (generalized) Splitting PAk-structures. Also PA1-chains and even PA1-chains of Lie-type can be constructed explicitly.


## 3. Applications of GAP-Theory on Solving Partial Differential Equations. New Explicit Solutions of Einstein Equations and Navier-Stokes Equations

In this chapter we will show, that GAP-Theory allows interesting applications on Theoretical Physics by calculating new explicit solutions of Einstein's field equations from General Relativity Theory and new explicit solutions of Navier-Stokes equations.In some cases, the gained solutions are presented not only in the (compact and elegant) notation of GAP-Theory but also in the more circumstantial classical notation. This allows the checking of the results by readers without knowledges in GAP-Theory (Assumed are only the definitions of the vector spaces𝔗1(W;*)(ξ0) and 𝔗1(W;*)(ξ0) from Section 2.1.) by inserting the presented solutions into the corresponding differential equations.


### 3.1. New Explicit Solutions of Einstein Equations

We start with Einsteins field equations(3.1)Rij-12Rgij=-κTij.
Here gij denotes the metric tensor of an n-dimensional Riemannian manifold 𝔐 with relativistic signature, Rij the Ricci tensor defined by
(3.2)Rij=gmkRmijk=12gmk(gmk,ij+gij,mk-gik,jm-gjm,ik)+gmkghl(ΓhijΓlmk-ΓhikΓlmj),Γhij=12(ghi,j-gij,h+gjh,i),Γhij the Christoffel symbols of first kind, and R:=gijRij the curvature scalar. The right side of Einstein equations is defined by the relativistic gravitational constant ϰ=8πγ/c4=1.86·10-27 cm/g (γ the classical gravitational constant) and the energy momentum tensor Tij of the considered matter.Einstein equations represent a system of nonlinear PDEs of second order to calculate the covariant componentsgij of the metric. The left side of these equations is linear in the second derivatives but nonlinear in the first derivatives of the covariant metric gij and also nonlinear in the covariant metric itself, because gmk is a nonlinear function of gij. Generally also the right side of Einstein equations is nonlinear in the metric and the type of nonlinearity depends on the concrete physically situation.In the past a lot of explicit solutions of Einstein equations have been constructed, but most of them are restricted on the very special caseRij=λgij (Einstein spaces). Among these solutions only a small set is of physical importance because most of them have too special mathematical symmetries for describing physically gravitational fields. By contrast we will develope a variety of explicit solutions in the framework of GAP-Theory without assuming Einstein spaces.Algebraic tools for solving Einstein equations have been used in [18, 19] (associative and commutative algebras) and [4] (nonassociative and noncommutative algebras; see also the overview given in [4]), discussions of certain generalizations of Einstein equations with algebra-valued metrics. Unlike we will deal here with the classical Einstein equations, that is, with real-valued, four dimensional metrics, and special GAPs are used for describing certain inherent symmetries of the explicit solutions. These GAPs differ from the algebras used in the papers cited above.Finally some remarks to the denotions in this chapter are presented:ηij is the covariante Minkowski metric, ηjk defined by ηijηjk=δjk. In relativistic theories, the indices run from 0 to 3 or—for generalizations—from 0 to n. In the framework of algebra theory, however, the left index range is not 0 but 1 and the relativistic denotion would lead to some problems and misunderstandings. For this we always will use the index range 1,…,4 or—for generalizations—from 1 to n. Furthermore, sometimes we will write 〈u,v〉 for gijuivj.Theorem 3.1 (in GAP-notation).
Given a four-dimensional splitting pseudoring(R4;*) with the splitting parameters  b,d,h and the restrictions
(3.3)b=1orb=-3,didjηij=0,
and further given an arbitrary vector field  a∈𝔗1(W;*)(ξ0),  W⊆𝔐, then all metric tensors of the form
(3.4)gij(ξ)=am(ξ)C(*)m{ilηj}l
(C(*) the corresponding structure constant tensor,  C(*)m{ilηj}l:=C(*)milηjl+C(*)mjlηil) are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system (ξ).Theorem 3.2 (in classical notation).
Given the realsb,di,hi,  i=1,…,4 with the restrictions
(3.5)b=1orb=-3,didjηij=0,dihi=-1,
And further given an arbitrary vector fielda∈𝔗1(W;*)(ξ0),  W⊆𝔐, then all metric tensors of the form
(3.6)gij(ξ)=2am(ξ)dmηij+b[diam(ξ)ηmj+djam(ξ)ηmi]+(am(ξ)dm)[dihlηlj+djhlηli]
are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system ( ξ).The proof is omitted for space reason, the interested reader may insert the solution into the vacuum Einstein equationsRij=0. It is remarkable that vacuum solutions are realizable in the case of commutative algebras (b=1) as well as in the case of noncommutative algebras (b=-3).Now we will show that the vacuum metric above is a new unknown metric and not only a certain representation of the well-known Kerr metric or if the metric in   [17]. For this we remember on the structure of the Kerr metric, which is given in Boyer-Lindquist coordinates t,r,ϕ,θ as follows:
(3.7)ds2=(1-rsrρ2)c2dt2+2rsrasin2θρ2cdtdϕ-ρ2Λ2dr2-ρ2dθ2-(r2+a2+rsra2ρ2sin2θ)dϕ2,rs the Schwarzschild radius, a the Kerr parameters and ρ2:=r2+a2cos2θ,  Λ2:=r2-rsr+a2. In the following matrix form:
(3.8)(gij)=((1-rsrρ2)02rsrasin2θρ2c0⋯-ρ2Λ200⋯⋯-(r2+a2+rsra2ρ2sin2θ)0⋯⋯⋯-ρ2),
remember that the two fundamental symmetries of the Kerr metric—which describes an uncharged rotating black hole—are given by (∂/∂t)gij=0 (time-independence) and (∂/∂ϕ)gij=0 (axial symmetry).The matrix form of our metric (3.6) is given by
(3.9)(gij)=(2(ψ+d1w1)d1w2+d2w1d1w3+d3w1d1w4+d4w1⋯2(-ψ+d2w2)d2w3+d3w2d2w4+d4w2⋯⋯2(-ψ+d3w3)d3w4+d4w3⋯⋯⋯2(-ψ+d4w4)),
with wj(ξ):=ηjk(bak+hk),ψ(ξ):=akdk,a∈𝔗1(W;*)(ξ0) an arbitrary field. It is easy to see that there does not exist any coordinate transformation such that the metric (3.9) takes the structure of the Kerr metric (3.8). The zeros in (3.8) cannot be realized by preserving the structure of the nontrivial elements, that is, our vacuum metric is not from Kerr type; the Kerr symmetries of time-independence and axial symmetry are replaced by the symmetry of *-analyticity.It remains to show that the metric is not from the type presented in   [17]. For this we compare the following:
(3.10)Metricin[17]:gij(ξ)=ηij+C(*){ij}lAl(ξ)=ηij+(C(*)ijl+C(*)jil)Al(ξ),Metricabove:gij(ξ)=am(ξ)(C(*)m{ilηj}l=am(ξ)(C(*)milηjl+C(*)mjlηil)
with A∈𝔗1(W;*)(ξ0),  a∈𝔗1(W;*)(ξ0). It is easy to see that the metrics are of different types.(i)
The intrinsic symmetries of the vector fieldsA∈𝔗1(W;*)(ξ0) and a∈𝔗1(W;*)(ξ0)  are of different types (𝔗1(W;*)(ξ0)≠𝔗1(W;*)(ξ0)).(ii)
The formal structure of the metrics differs, because the structure constant tensor appears in different manners:(C(*)ijlAl(ξ)≠am(ξ)C(*)mil,C(*){ij}l≠C(*)m{ilηj}l).(iii)
The metric in   [17] cannot describe nontrivial vacuum solutions (i.e., vacuum solutions for nonflat spaces; see   [17]), whereas our new metric describes nontrivial vacuum solutions.For these reasons, we recognize the metric presented here as a new, unknown vacuum metric. Now we will solve Einstein equations for a nonvanishing stress energy tensor.Theorem 3.3 (in GAP-notation).
Given ann-dimensional splitting pseudoring  (ℝn;*) with the splitting parameters  b,d,hand the restrictions
(3.11)didjηij=0,
and further given the  n-dimensional Einstein equations in a local coordinate system  (ξ) in the form
(3.12)Rij-12Rgij=-ϰεdidjwithε(ξ)=-ϰ-1b2(n-2)〈w,𝔇(*)2f〉-b〈w,𝔇(*)f〉〈d,𝔇(*)a〉2(1+〈w,𝔇(*)f〉),wldl=0,a,f∈𝔗1(W;*)(ξ0), W⊆𝔐arbitrary vector fields, restricted only by the condition ai≠γdi,  γ a scalar field, wlconstants, then these equations can be solved explicitly by
(3.13)gij(ξ)=(1+〈w,𝔇(*)f〉)ηij+(C(*)ijh+C(*)jih)ah(ξ),C(*)the structure constant tensor of the splitting pseudoring.Theorem 3.4 (in classical notation).
Given the realsb,di,hi,wi,i=1,…,nwith the restrictions
(3.14)didjηij=0,hidi=-1,wldl=0,
and further given the n-dimensional Einstein equations in a local coordinate system  (ξ) in the form
(3.15)Rij-12Rgij=-ϰε(ξ)didj,εa scalar field, these equations can be solved explicitly by
(3.16)gij(ξ)=(1+wm(𝔇(*)f)m)ηij+(b+1)[diaj(ξ)+djai(ξ)]+2didjhmam(ξ),
where C(*)isthe structure constant tensor of the splitting pseudoring,  a,f∈𝔗1(W;*)(ξ0),  W⊆𝔐arbitrary vector fields, restricted only by the condition ai≠γdi,  γa scalar field. The concrete structure of the vector fields  a,fdefines the structure of the scalar field ε.The proof is omitted for space reason, the interested reader may insert the solution above into the Einstein equations. Theorem3.4 is less general than Theorem 3.3 because nothing is told about the concrete structure of the scalar field ε. (The classical notation of ε would be very circumstantial.) We see that solutions of the inhomogeneous n-dimensional Einstein equations are realizable in the case of commutative algebras (b=1) as well as in the case of noncommutative algebras (b≠1), and that a noncommutative algebra with b=-1 exists, where the solutions take the most simple structure.Consider, furthermore, that Theorem3.3 is not a simple generalization of Theorem 3.1 by generalizing the dimension and introducing a nonvanishing stress energy tensor. Theorem 3.1 works in the framework of 𝔗1(W;*)(ξ0), whereas Theorem 3.3 works in the framework of 𝔗1(W;*)(ξ0).It remains to show that the metric in Theorem3.3 differs from the metric presented in   [17]. For this we compare the following:
(3.17)Metricin[17]:gij(ξ)=ηij+(C(*)ijl+C(*)jil)Al(ξ),witharbitrarysplittingparametersMetricabove:gij(ξ)=(1+〈w,𝔇(*)f〉)ηij+(C(*)ijl+C(*)jil)al(ξ)withrestrictedsplittingparametersdidjηij=0,
with A,a,f∈𝔗1(W;*)(ξ0). The metrics become identical if we restrict the algebraic framework of the first metric by didjηij=0 and the second metric by 𝔇(*)f=0; in all other cases the metrics are different.Finally some remarks to the physical importance of the stress energy tensor above are presented. For this we have a look on the following overview:(3.18a)stressenergytensorphysicalcharacterizationTij=(p+ε)uiuj+pgijmacroskopicmatter(3.18b)Tij=14π(FimFjm-14FpqFpqgij)withFim:=Am;i-Ai;mfreeelectromagneticstressenergytensor(3.18c)Tij=μ0uiuj+14π(FimFjm-14FpqFpqgij)incoherentmatterintheelectromagneticfield.(i)
The case (3.18a) describes macroscopic matter with velocity ui, pressure p, and energy density ε.(ii)
The case (3.18b) describes the interchanging of the space-time structure with the electromagnetic field, where F denotes the antisymmetric electromagnetic field tensor, defined by the potential A.(iii)
The case in (3.18c) describes the interchanging of the space-time structure with incoherent matter in the electromagnetic field.The stress energy tensor considered in Theorem3.3 is defined by Tij=εdidj, which allows a physical interpretation as macroscopic matter with velocity di (constancy by introducing a special coordinate system), energy density ε, and pressure p=0. It can be shown that also special electromagnetic fields lead to the stress energy tensor above.


### 3.2. New Explicit Solutions of Navier-Stokes Equations

“Although these [Navier-Stokes] equations were written down in the 19th century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.”With these words, the Clay Mathematic Institute describes a problem which is denoted as one of the seven “Millennium Problems” (see athttp://www.claymath.org/). In the official formulation (The complete formulation can be found for instance in http://www.claymath.org./) by   [20], the Navier-Stokes equations are presented as follows.“The Euler and Navier-Stokes equations describe the motion of a fluid inℝn(n=2 or 3). These equations are to be solved for an unknown velocity vector u(x,t)=(ui(x,t))1≤i≤n∈ℝn and pressure p(x,t)∈ℝ, defined for position x∈ℝn and time t≥0. We restrict attention here to incompressible fluids filling all of ℝn. The Navier-Stokes equations are then given by(3.19a)∂∂tui+∑j=1nuj∂ui∂xj‍=νΔui-∂p∂xi+fi(x,t),x∈ℝn,t≥0,(3.19b)divu=∑j=1n∂ui∂xi=0‍,x∈ℝn,t≥0,
with initial conditions
(3.20)u(x,0)=u0(x),x∈ℝn.Here,u0(x) is a given, C∞ divergence-free vector field on ℝn,fi(x,t) are the components of a given, externally applied force (e.g., gravity), ν is a positive coefficient (the viscosity), and Δ=∑j=1n(∂2/∂xj2)‍  is the Laplacian in the space variables…”Before presenting our new GAP-solution method, we remember some standard methods and main results of Navier-Stokes solution theory. Leray   [21] has introduced the program of analyzing weak solutions of Navier-Stokes equations and he was able to show that in the three-dimensional case always exists a weak solution with suitable growth properties. Lerays method was taken and amended by Scheffer   [22], Caffarelli et al.   [23], Lin [24], Shnirelman   [25], and others with the main result that the growth conditions of weak solutions can be classified quantitative [23, 24].But at time it is not possible to perform the step from weak solutions to the physically interesting smooth solutions; we do not know if smooth initial conditions always induce smooth solutions or not. Also uniqueness of weak solutions is not known. Furthermore, the concrete construction of Navier-Stokes-solutions is a very hard problem. Standard methods for a nonnumerical construction are group methods, also the classical method of separation is used in a modified way, but actually the gained results are very small, which is not a surprise; all  nonnumerical methods deal with very symmetric structures, and we cannot expect that such special tools delivers very general results.In the official formulation   [20] of the “Navier-Stokes Millenium Problem”, these difficulties are summarized as follows:“⋯our understand [of the Navier-Stokes equations] is a very primitive level.  Standard methods from PDE appear inadequate to settle the problem.”In the following, we will rewrite the Navier-Stokes system (3.19a) and (3.19b). With  (In the following considerations, always a cartesian coordinate system is assumed. Thus the contravariant components ui of the vector field u can be identified with the covariant components ui from   [20] in (3.19a) and (3.19b)). u,ji:=∂ui/∂xj,  qi:=-p,i+fi, the Einstein summation convention, and the generalization of ℝn to a region W⊆ℝn (in the case W≠ℝn, adequate boundary conditions must be assumed), we get(3.21a)u,ti-νΔui+uju,ji=qi,(3.21b)u,ii=0,forx∈W,t≥0.
We will solve these equations under the assumptions u(x,t)=U(z(x,t)),  q(x,t)=Q(z(x,t)), where the n-dimensional vector fields U(z) and Q(z) are assumed as *-analytic in the n-dimensional vector z of the following structure:
(3.22)z(x,t)=λ*x+μt,W⊆ℝn,t∈[0,∞[,z∈Wz.
Here λ,μ∈ℝn denote arbitrary fixed constants and Wz⊆ℝn denotes the region of z.Theorem 3.5.
Given a commutative pseudoring(ℝn;*)with structure constants C(*)ijk,C(*)[3]m:=C(*)iim,  C(*)[1]m:=C(*)mii, and further given two arbitrary vectors λ,μ∈ℝn, the n-dimensional Navier-Stokes system (3.21a) and (3.21b) with viscosity coefficientν∈ℝand an arbitrary inhomogeneity of the symmetry
(3.23)q(x,t)=Q(z(x,t)),Q∈𝔗1(Wz;*)(z0)an arbitrary field
with z(x,t) from (3.22), then the following statements hold.Statement 1.
A solutionu(x,t) of the Navier-Stokes differential equations  (3.21a) is given by
(3.24)(a)u(x,t)=U(z(x,t)),(b)U(z)=-2νλ*C(*)[3]*Y-1(*)(z)*𝔇(*)Y,Y(z)definedbythelineardifferentialequation,(c)ν2(C(*)[3])2*λ3*𝔇(*)2Y-νC(*)[3]*λ*μ*𝔇(*)Y-12(𝔇(*)-1Q)*Y=0.Statement 2.
If the inhomogeneityQ satisfies the condition
(3.25)C(*)i[jk](𝔇(*)Q)i=0,
it becomes interpretable as the pressure gradient  Q=gradp, and therefore the pressure is explicitly given by
(3.26)p(x,t)=∫Qi(z(x,t))dxi‍.Statement 3.
If furthermore(3.27)C(*)[1]iC(*)ijkλj=0,
also the divergence condition (3.21b) is satisfied.Before considering the proof structure (the extensive proof is omitted for space reason), we will give some hints to the statements above from a physical point of view as well as from a mathematical point of view. We begin by remembering that the entire Navier-Stokes-problem as formulated in (3.19a) and (3.19b) can be split up to three subproblems:(1)
the problem of solving the Navier-Stokes differential equation (3.19a) for an arbitrary given inhomogeneity (-∂p/∂xi+fi must not be a gradient),(2)
the problem of restricting (3.19a) solutions by demanding inhomogeneities of gradient form.(3)
the problem of restricting (3.19a)-solutions (for gradient inhomogenities as well as for arbitrary inhomogeneities) by demanding the divergence condition (3.19b).The solution of problem 1 is interesting from a mathematical point of view and the solution of problem 2 is interesting not only from a mathematical but also from a physical point of view, because for many practical applications, gradient inhomogeneities (pressure gradient) are assumed. Finally the solution of problem 3 (for gradient inhomogeneities) is interesting for getting a prize (see the “Millennium Problem”).Our statements in the Theorem3.5 follow this classification: Statement 1 solves the problem 1 (without any restriction on the used algebraic framework), Statement 2 solves the problem 2 (by restricting the algebraic framework according to (3.25)), and Statement 3 gives a condition for solving the remaining problem 3 (by a further restriction of the algebraic framework according to (3.27)). It can be shown explicitly that algebras exist, such that the problems 1 and 2 can be solved explicitly (Kronecker-algebras, see the Section 2), but at time we do not know an algebra such that all three problems become solvable at the same time (for a new, unknown solution set).Therefore, the statements of Theorem3.5 are interesting from a mathematical point of view (Statement 1) as well as from a physical point of view (Statement 2), but—of course—the Millennium Problem remains unsolved.Now some remarks to the solution structure (3.21a) and (3.21b) are presented.(i)
The solution structure holds for arbitrary commutative pseudorings; we are not restricted on splitting pseudorings as in the case of the Einstein solutions above.(ii)
The vectorY-1(*)(z) appears, that is, the solution U(z)generally will have singularities. In division algebras, these singularities are described entirely by values zα, α=1,2,… with the property Y-1(*)(zα)=0, but in nondivision algebras there exist a larger set of singularities (see Section 2 and also the following considerations). In other words, the solution structure above allows an explicit description of the singularities depending on the chosen algebra.(iii)
It must be mentioned that our considerations here were restricted on the simple cases of commutative pseudorings and the special𝔗1-symmetry. Noncommutative pseudorings and generalizations of 𝔗1-symmetry (or 𝔗1-symmetry) lead to very complex Navier-Stokes solutions which cannot be presented in this framework.Finally some hints to the proof-structure are presented.(1)
It is shown that in the case of a commutative pseudoring (which is assumed) the assumed symmetryQ(z)∈𝔗1(Wz;*)(z0) of the inhomogeneity leads to the symmetry u(x,t)=U(z)∈𝔗1(Wz;*)(z0).(2)
This property ofU now allows the transformation of the Navier-Stokes differential equation (3.21a) onto a ”GAP-Riccati differential equation” (i.e., an algebra valued Riccati equation).(3)
The assumed commutativity of the pseudoring now allows to solve this GAP-Riccati equation in the same way as the classical scalar Riccati-equation by transforming this equation onto a linear (algebra-valued) second-order differential equation (see (3.24)(c)).(4)
The explicit solution of this linear differential equation by comparison of coefficients is a simple problem, which only depends on the concrete structure of𝔇(*)-1Q, that is, on the structure of the given inhomogeneity Q.(5)
After calculating the vector fieldY(z) from (3.24)(c), we get the vector field U(z) according to (3.24)(b). The inverse vector field Y-1(*)(z) is calculable explicitly for the physical interesting dimensions n≤3, in the case of a splitting pseudoring (see Section 2) also for arbitrary dimensions n, and therefore, U(z) is calculable explicitly.(6)
After calculating the vector fieldU(z), we get the Navier-Stokes solution u(x,t) according to (3.24)(a) by inserting z(x,t)=λ*x+μt (see (3.22)).


### 3.3. The Necessity of an Embedding Formalism

The results above show that GAP-Theory allows the explicit solution of important nonlinear partial differential equations from Theoretical Physics. For space reason, we have restricted these applications on Einstein field equations and Navier-Stokes equations, but GAP-Theory also allows applications on other famous differential equations as the Yang-Mills equation, the Korteweg-de Vries equation, and so forth.The fundamental symmetry-framework of our considerations was the symmetry of*-analyticity (GAP-symmetry) which has turned out as “compatible” with the considered partial differential equations in the sense that it allows an explicit solution formalism. From the mathematical point of view, *-analyticity seems to be a useful tool to look inside the structure of covariant differential equations, but from the physical point of view *-analyticity represents a strong restriction: in many cases problem defining physically vector fields (tensor fields) will be not be *-analytical but only smooth.To demonstrate the concrete problems, we have a look on the Theorem3.5 for solving the Navier-Stokes system (3.21a) and (3.21b) in the framework of *-analyticity: Theorem 3.5 shows that a *-analytical inhomogeneity q allows an explicit solution of the Navier-Stokes differential equation (3.21a) for arbitrary commutative pseudorings, and that furthermore for a special set of these pseudorings also the divergence condition (3.21b) holds. But for many applications, the inhomogeneity q will not be *-analytical but a gradient, and in this case Theorem 3.5 is applicable only on *-analytical gradients, which represents a strong restriction.Similar considerations hold for the Einstein equations: In Theorem3.3  *-analyticity appears in the stress energy tensor (in a hidden form) as well as in the metric tensor, but a lot of physical interesting stress energy tensors will not be *-analytical.These considerations boil down to the following question: is it possible to embed an arbitraryn-dimensional smooth vector field f (tensor field) into an N-dimensional *-analytical vector field F (tensor field) for a sufficient large dimension N≥n in the sense, that the first n components of F allow a representation of the n components of f? (An exact definition of “embedding” will be given in Section 4.)In the case of a positive answer,we can embed an-dimensional nonlinear partial differential equation defined by only smooth vector fields onto aN-dimensional nonlinear partial differential equation defined by*-analytical vector fields, and this high-dimensional differential equation can be solved in the framework of GAP-Theory as demonstrated in the sections above.Finally the firstn components of the obtained N-dimensional solution vector field (tensor field) represent the solution of the n-dimensional start problem.Section4 will show that such an Embedding formalism of smoothness into *-analyticity (or more general: chain analyticity) exists.


## 3.1. New Explicit Solutions of Einstein Equations

We start with Einsteins field equations(3.1)Rij-12Rgij=-κTij.
Here gij denotes the metric tensor of an n-dimensional Riemannian manifold 𝔐 with relativistic signature, Rij the Ricci tensor defined by
(3.2)Rij=gmkRmijk=12gmk(gmk,ij+gij,mk-gik,jm-gjm,ik)+gmkghl(ΓhijΓlmk-ΓhikΓlmj),Γhij=12(ghi,j-gij,h+gjh,i),Γhij the Christoffel symbols of first kind, and R:=gijRij the curvature scalar. The right side of Einstein equations is defined by the relativistic gravitational constant ϰ=8πγ/c4=1.86·10-27 cm/g (γ the classical gravitational constant) and the energy momentum tensor Tij of the considered matter.Einstein equations represent a system of nonlinear PDEs of second order to calculate the covariant componentsgij of the metric. The left side of these equations is linear in the second derivatives but nonlinear in the first derivatives of the covariant metric gij and also nonlinear in the covariant metric itself, because gmk is a nonlinear function of gij. Generally also the right side of Einstein equations is nonlinear in the metric and the type of nonlinearity depends on the concrete physically situation.In the past a lot of explicit solutions of Einstein equations have been constructed, but most of them are restricted on the very special caseRij=λgij (Einstein spaces). Among these solutions only a small set is of physical importance because most of them have too special mathematical symmetries for describing physically gravitational fields. By contrast we will develope a variety of explicit solutions in the framework of GAP-Theory without assuming Einstein spaces.Algebraic tools for solving Einstein equations have been used in [18, 19] (associative and commutative algebras) and [4] (nonassociative and noncommutative algebras; see also the overview given in [4]), discussions of certain generalizations of Einstein equations with algebra-valued metrics. Unlike we will deal here with the classical Einstein equations, that is, with real-valued, four dimensional metrics, and special GAPs are used for describing certain inherent symmetries of the explicit solutions. These GAPs differ from the algebras used in the papers cited above.Finally some remarks to the denotions in this chapter are presented:ηij is the covariante Minkowski metric, ηjk defined by ηijηjk=δjk. In relativistic theories, the indices run from 0 to 3 or—for generalizations—from 0 to n. In the framework of algebra theory, however, the left index range is not 0 but 1 and the relativistic denotion would lead to some problems and misunderstandings. For this we always will use the index range 1,…,4 or—for generalizations—from 1 to n. Furthermore, sometimes we will write 〈u,v〉 for gijuivj.Theorem 3.1 (in GAP-notation).
Given a four-dimensional splitting pseudoring(R4;*) with the splitting parameters  b,d,h and the restrictions
(3.3)b=1orb=-3,didjηij=0,
and further given an arbitrary vector field  a∈𝔗1(W;*)(ξ0),  W⊆𝔐, then all metric tensors of the form
(3.4)gij(ξ)=am(ξ)C(*)m{ilηj}l
(C(*) the corresponding structure constant tensor,  C(*)m{ilηj}l:=C(*)milηjl+C(*)mjlηil) are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system (ξ).Theorem 3.2 (in classical notation).
Given the realsb,di,hi,  i=1,…,4 with the restrictions
(3.5)b=1orb=-3,didjηij=0,dihi=-1,
And further given an arbitrary vector fielda∈𝔗1(W;*)(ξ0),  W⊆𝔐, then all metric tensors of the form
(3.6)gij(ξ)=2am(ξ)dmηij+b[diam(ξ)ηmj+djam(ξ)ηmi]+(am(ξ)dm)[dihlηlj+djhlηli]
are solutions of the four-dimensional vacuum Einstein equations in a local coordinate system ( ξ).The proof is omitted for space reason, the interested reader may insert the solution into the vacuum Einstein equationsRij=0. It is remarkable that vacuum solutions are realizable in the case of commutative algebras (b=1) as well as in the case of noncommutative algebras (b=-3).Now we will show that the vacuum metric above is a new unknown metric and not only a certain representation of the well-known Kerr metric or if the metric in   [17]. For this we remember on the structure of the Kerr metric, which is given in Boyer-Lindquist coordinates t,r,ϕ,θ as follows:
(3.7)ds2=(1-rsrρ2)c2dt2+2rsrasin2θρ2cdtdϕ-ρ2Λ2dr2-ρ2dθ2-(r2+a2+rsra2ρ2sin2θ)dϕ2,rs the Schwarzschild radius, a the Kerr parameters and ρ2:=r2+a2cos2θ,  Λ2:=r2-rsr+a2. In the following matrix form:
(3.8)(gij)=((1-rsrρ2)02rsrasin2θρ2c0⋯-ρ2Λ200⋯⋯-(r2+a2+rsra2ρ2sin2θ)0⋯⋯⋯-ρ2),
remember that the two fundamental symmetries of the Kerr metric—which describes an uncharged rotating black hole—are given by (∂/∂t)gij=0 (time-independence) and (∂/∂ϕ)gij=0 (axial symmetry).The matrix form of our metric (3.6) is given by
(3.9)(gij)=(2(ψ+d1w1)d1w2+d2w1d1w3+d3w1d1w4+d4w1⋯2(-ψ+d2w2)d2w3+d3w2d2w4+d4w2⋯⋯2(-ψ+d3w3)d3w4+d4w3⋯⋯⋯2(-ψ+d4w4)),
with wj(ξ):=ηjk(bak+hk),ψ(ξ):=akdk,a∈𝔗1(W;*)(ξ0) an arbitrary field. It is easy to see that there does not exist any coordinate transformation such that the metric (3.9) takes the structure of the Kerr metric (3.8). The zeros in (3.8) cannot be realized by preserving the structure of the nontrivial elements, that is, our vacuum metric is not from Kerr type; the Kerr symmetries of time-independence and axial symmetry are replaced by the symmetry of *-analyticity.It remains to show that the metric is not from the type presented in   [17]. For this we compare the following:
(3.10)Metricin[17]:gij(ξ)=ηij+C(*){ij}lAl(ξ)=ηij+(C(*)ijl+C(*)jil)Al(ξ),Metricabove:gij(ξ)=am(ξ)(C(*)m{ilηj}l=am(ξ)(C(*)milηjl+C(*)mjlηil)
with A∈𝔗1(W;*)(ξ0),  a∈𝔗1(W;*)(ξ0). It is easy to see that the metrics are of different types.(i)
The intrinsic symmetries of the vector fieldsA∈𝔗1(W;*)(ξ0) and a∈𝔗1(W;*)(ξ0)  are of different types (𝔗1(W;*)(ξ0)≠𝔗1(W;*)(ξ0)).(ii)
The formal structure of the metrics differs, because the structure constant tensor appears in different manners:(C(*)ijlAl(ξ)≠am(ξ)C(*)mil,C(*){ij}l≠C(*)m{ilηj}l).(iii)
The metric in   [17] cannot describe nontrivial vacuum solutions (i.e., vacuum solutions for nonflat spaces; see   [17]), whereas our new metric describes nontrivial vacuum solutions.For these reasons, we recognize the metric presented here as a new, unknown vacuum metric. Now we will solve Einstein equations for a nonvanishing stress energy tensor.Theorem 3.3 (in GAP-notation).
Given ann-dimensional splitting pseudoring  (ℝn;*) with the splitting parameters  b,d,hand the restrictions
(3.11)didjηij=0,
and further given the  n-dimensional Einstein equations in a local coordinate system  (ξ) in the form
(3.12)Rij-12Rgij=-ϰεdidjwithε(ξ)=-ϰ-1b2(n-2)〈w,𝔇(*)2f〉-b〈w,𝔇(*)f〉〈d,𝔇(*)a〉2(1+〈w,𝔇(*)f〉),wldl=0,a,f∈𝔗1(W;*)(ξ0), W⊆𝔐arbitrary vector fields, restricted only by the condition ai≠γdi,  γ a scalar field, wlconstants, then these equations can be solved explicitly by
(3.13)gij(ξ)=(1+〈w,𝔇(*)f〉)ηij+(C(*)ijh+C(*)jih)ah(ξ),C(*)the structure constant tensor of the splitting pseudoring.Theorem 3.4 (in classical notation).
Given the realsb,di,hi,wi,i=1,…,nwith the restrictions
(3.14)didjηij=0,hidi=-1,wldl=0,
and further given the n-dimensional Einstein equations in a local coordinate system  (ξ) in the form
(3.15)Rij-12Rgij=-ϰε(ξ)didj,εa scalar field, these equations can be solved explicitly by
(3.16)gij(ξ)=(1+wm(𝔇(*)f)m)ηij+(b+1)[diaj(ξ)+djai(ξ)]+2didjhmam(ξ),
where C(*)isthe structure constant tensor of the splitting pseudoring,  a,f∈𝔗1(W;*)(ξ0),  W⊆𝔐arbitrary vector fields, restricted only by the condition ai≠γdi,  γa scalar field. The concrete structure of the vector fields  a,fdefines the structure of the scalar field ε.The proof is omitted for space reason, the interested reader may insert the solution above into the Einstein equations. Theorem3.4 is less general than Theorem 3.3 because nothing is told about the concrete structure of the scalar field ε. (The classical notation of ε would be very circumstantial.) We see that solutions of the inhomogeneous n-dimensional Einstein equations are realizable in the case of commutative algebras (b=1) as well as in the case of noncommutative algebras (b≠1), and that a noncommutative algebra with b=-1 exists, where the solutions take the most simple structure.Consider, furthermore, that Theorem3.3 is not a simple generalization of Theorem 3.1 by generalizing the dimension and introducing a nonvanishing stress energy tensor. Theorem 3.1 works in the framework of 𝔗1(W;*)(ξ0), whereas Theorem 3.3 works in the framework of 𝔗1(W;*)(ξ0).It remains to show that the metric in Theorem3.3 differs from the metric presented in   [17]. For this we compare the following:
(3.17)Metricin[17]:gij(ξ)=ηij+(C(*)ijl+C(*)jil)Al(ξ),witharbitrarysplittingparametersMetricabove:gij(ξ)=(1+〈w,𝔇(*)f〉)ηij+(C(*)ijl+C(*)jil)al(ξ)withrestrictedsplittingparametersdidjηij=0,
with A,a,f∈𝔗1(W;*)(ξ0). The metrics become identical if we restrict the algebraic framework of the first metric by didjηij=0 and the second metric by 𝔇(*)f=0; in all other cases the metrics are different.Finally some remarks to the physical importance of the stress energy tensor above are presented. For this we have a look on the following overview:(3.18a)stressenergytensorphysicalcharacterizationTij=(p+ε)uiuj+pgijmacroskopicmatter(3.18b)Tij=14π(FimFjm-14FpqFpqgij)withFim:=Am;i-Ai;mfreeelectromagneticstressenergytensor(3.18c)Tij=μ0uiuj+14π(FimFjm-14FpqFpqgij)incoherentmatterintheelectromagneticfield.(i)
The case (3.18a) describes macroscopic matter with velocity ui, pressure p, and energy density ε.(ii)
The case (3.18b) describes the interchanging of the space-time structure with the electromagnetic field, where F denotes the antisymmetric electromagnetic field tensor, defined by the potential A.(iii)
The case in (3.18c) describes the interchanging of the space-time structure with incoherent matter in the electromagnetic field.The stress energy tensor considered in Theorem3.3 is defined by Tij=εdidj, which allows a physical interpretation as macroscopic matter with velocity di (constancy by introducing a special coordinate system), energy density ε, and pressure p=0. It can be shown that also special electromagnetic fields lead to the stress energy tensor above.


## 3.2. New Explicit Solutions of Navier-Stokes Equations

“Although these [Navier-Stokes] equations were written down in the 19th century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.”With these words, the Clay Mathematic Institute describes a problem which is denoted as one of the seven “Millennium Problems” (see athttp://www.claymath.org/). In the official formulation (The complete formulation can be found for instance in http://www.claymath.org./) by   [20], the Navier-Stokes equations are presented as follows.“The Euler and Navier-Stokes equations describe the motion of a fluid inℝn(n=2 or 3). These equations are to be solved for an unknown velocity vector u(x,t)=(ui(x,t))1≤i≤n∈ℝn and pressure p(x,t)∈ℝ, defined for position x∈ℝn and time t≥0. We restrict attention here to incompressible fluids filling all of ℝn. The Navier-Stokes equations are then given by(3.19a)∂∂tui+∑j=1nuj∂ui∂xj‍=νΔui-∂p∂xi+fi(x,t),x∈ℝn,t≥0,(3.19b)divu=∑j=1n∂ui∂xi=0‍,x∈ℝn,t≥0,
with initial conditions
(3.20)u(x,0)=u0(x),x∈ℝn.Here,u0(x) is a given, C∞ divergence-free vector field on ℝn,fi(x,t) are the components of a given, externally applied force (e.g., gravity), ν is a positive coefficient (the viscosity), and Δ=∑j=1n(∂2/∂xj2)‍  is the Laplacian in the space variables…”Before presenting our new GAP-solution method, we remember some standard methods and main results of Navier-Stokes solution theory. Leray   [21] has introduced the program of analyzing weak solutions of Navier-Stokes equations and he was able to show that in the three-dimensional case always exists a weak solution with suitable growth properties. Lerays method was taken and amended by Scheffer   [22], Caffarelli et al.   [23], Lin [24], Shnirelman   [25], and others with the main result that the growth conditions of weak solutions can be classified quantitative [23, 24].But at time it is not possible to perform the step from weak solutions to the physically interesting smooth solutions; we do not know if smooth initial conditions always induce smooth solutions or not. Also uniqueness of weak solutions is not known. Furthermore, the concrete construction of Navier-Stokes-solutions is a very hard problem. Standard methods for a nonnumerical construction are group methods, also the classical method of separation is used in a modified way, but actually the gained results are very small, which is not a surprise; all  nonnumerical methods deal with very symmetric structures, and we cannot expect that such special tools delivers very general results.In the official formulation   [20] of the “Navier-Stokes Millenium Problem”, these difficulties are summarized as follows:“⋯our understand [of the Navier-Stokes equations] is a very primitive level.  Standard methods from PDE appear inadequate to settle the problem.”In the following, we will rewrite the Navier-Stokes system (3.19a) and (3.19b). With  (In the following considerations, always a cartesian coordinate system is assumed. Thus the contravariant components ui of the vector field u can be identified with the covariant components ui from   [20] in (3.19a) and (3.19b)). u,ji:=∂ui/∂xj,  qi:=-p,i+fi, the Einstein summation convention, and the generalization of ℝn to a region W⊆ℝn (in the case W≠ℝn, adequate boundary conditions must be assumed), we get(3.21a)u,ti-νΔui+uju,ji=qi,(3.21b)u,ii=0,forx∈W,t≥0.
We will solve these equations under the assumptions u(x,t)=U(z(x,t)),  q(x,t)=Q(z(x,t)), where the n-dimensional vector fields U(z) and Q(z) are assumed as *-analytic in the n-dimensional vector z of the following structure:
(3.22)z(x,t)=λ*x+μt,W⊆ℝn,t∈[0,∞[,z∈Wz.
Here λ,μ∈ℝn denote arbitrary fixed constants and Wz⊆ℝn denotes the region of z.Theorem 3.5.
Given a commutative pseudoring(ℝn;*)with structure constants C(*)ijk,C(*)[3]m:=C(*)iim,  C(*)[1]m:=C(*)mii, and further given two arbitrary vectors λ,μ∈ℝn, the n-dimensional Navier-Stokes system (3.21a) and (3.21b) with viscosity coefficientν∈ℝand an arbitrary inhomogeneity of the symmetry
(3.23)q(x,t)=Q(z(x,t)),Q∈𝔗1(Wz;*)(z0)an arbitrary field
with z(x,t) from (3.22), then the following statements hold.Statement 1.
A solutionu(x,t) of the Navier-Stokes differential equations  (3.21a) is given by
(3.24)(a)u(x,t)=U(z(x,t)),(b)U(z)=-2νλ*C(*)[3]*Y-1(*)(z)*𝔇(*)Y,Y(z)definedbythelineardifferentialequation,(c)ν2(C(*)[3])2*λ3*𝔇(*)2Y-νC(*)[3]*λ*μ*𝔇(*)Y-12(𝔇(*)-1Q)*Y=0.Statement 2.
If the inhomogeneityQ satisfies the condition
(3.25)C(*)i[jk](𝔇(*)Q)i=0,
it becomes interpretable as the pressure gradient  Q=gradp, and therefore the pressure is explicitly given by
(3.26)p(x,t)=∫Qi(z(x,t))dxi‍.Statement 3.
If furthermore(3.27)C(*)[1]iC(*)ijkλj=0,
also the divergence condition (3.21b) is satisfied.Before considering the proof structure (the extensive proof is omitted for space reason), we will give some hints to the statements above from a physical point of view as well as from a mathematical point of view. We begin by remembering that the entire Navier-Stokes-problem as formulated in (3.19a) and (3.19b) can be split up to three subproblems:(1)
the problem of solving the Navier-Stokes differential equation (3.19a) for an arbitrary given inhomogeneity (-∂p/∂xi+fi must not be a gradient),(2)
the problem of restricting (3.19a) solutions by demanding inhomogeneities of gradient form.(3)
the problem of restricting (3.19a)-solutions (for gradient inhomogenities as well as for arbitrary inhomogeneities) by demanding the divergence condition (3.19b).The solution of problem 1 is interesting from a mathematical point of view and the solution of problem 2 is interesting not only from a mathematical but also from a physical point of view, because for many practical applications, gradient inhomogeneities (pressure gradient) are assumed. Finally the solution of problem 3 (for gradient inhomogeneities) is interesting for getting a prize (see the “Millennium Problem”).Our statements in the Theorem3.5 follow this classification: Statement 1 solves the problem 1 (without any restriction on the used algebraic framework), Statement 2 solves the problem 2 (by restricting the algebraic framework according to (3.25)), and Statement 3 gives a condition for solving the remaining problem 3 (by a further restriction of the algebraic framework according to (3.27)). It can be shown explicitly that algebras exist, such that the problems 1 and 2 can be solved explicitly (Kronecker-algebras, see the Section 2), but at time we do not know an algebra such that all three problems become solvable at the same time (for a new, unknown solution set).Therefore, the statements of Theorem3.5 are interesting from a mathematical point of view (Statement 1) as well as from a physical point of view (Statement 2), but—of course—the Millennium Problem remains unsolved.Now some remarks to the solution structure (3.21a) and (3.21b) are presented.(i)
The solution structure holds for arbitrary commutative pseudorings; we are not restricted on splitting pseudorings as in the case of the Einstein solutions above.(ii)
The vectorY-1(*)(z) appears, that is, the solution U(z)generally will have singularities. In division algebras, these singularities are described entirely by values zα, α=1,2,… with the property Y-1(*)(zα)=0, but in nondivision algebras there exist a larger set of singularities (see Section 2 and also the following considerations). In other words, the solution structure above allows an explicit description of the singularities depending on the chosen algebra.(iii)
It must be mentioned that our considerations here were restricted on the simple cases of commutative pseudorings and the special𝔗1-symmetry. Noncommutative pseudorings and generalizations of 𝔗1-symmetry (or 𝔗1-symmetry) lead to very complex Navier-Stokes solutions which cannot be presented in this framework.Finally some hints to the proof-structure are presented.(1)
It is shown that in the case of a commutative pseudoring (which is assumed) the assumed symmetryQ(z)∈𝔗1(Wz;*)(z0) of the inhomogeneity leads to the symmetry u(x,t)=U(z)∈𝔗1(Wz;*)(z0).(2)
This property ofU now allows the transformation of the Navier-Stokes differential equation (3.21a) onto a ”GAP-Riccati differential equation” (i.e., an algebra valued Riccati equation).(3)
The assumed commutativity of the pseudoring now allows to solve this GAP-Riccati equation in the same way as the classical scalar Riccati-equation by transforming this equation onto a linear (algebra-valued) second-order differential equation (see (3.24)(c)).(4)
The explicit solution of this linear differential equation by comparison of coefficients is a simple problem, which only depends on the concrete structure of𝔇(*)-1Q, that is, on the structure of the given inhomogeneity Q.(5)
After calculating the vector fieldY(z) from (3.24)(c), we get the vector field U(z) according to (3.24)(b). The inverse vector field Y-1(*)(z) is calculable explicitly for the physical interesting dimensions n≤3, in the case of a splitting pseudoring (see Section 2) also for arbitrary dimensions n, and therefore, U(z) is calculable explicitly.(6)
After calculating the vector fieldU(z), we get the Navier-Stokes solution u(x,t) according to (3.24)(a) by inserting z(x,t)=λ*x+μt (see (3.22)).


## 3.3. The Necessity of an Embedding Formalism

The results above show that GAP-Theory allows the explicit solution of important nonlinear partial differential equations from Theoretical Physics. For space reason, we have restricted these applications on Einstein field equations and Navier-Stokes equations, but GAP-Theory also allows applications on other famous differential equations as the Yang-Mills equation, the Korteweg-de Vries equation, and so forth.The fundamental symmetry-framework of our considerations was the symmetry of*-analyticity (GAP-symmetry) which has turned out as “compatible” with the considered partial differential equations in the sense that it allows an explicit solution formalism. From the mathematical point of view, *-analyticity seems to be a useful tool to look inside the structure of covariant differential equations, but from the physical point of view *-analyticity represents a strong restriction: in many cases problem defining physically vector fields (tensor fields) will be not be *-analytical but only smooth.To demonstrate the concrete problems, we have a look on the Theorem3.5 for solving the Navier-Stokes system (3.21a) and (3.21b) in the framework of *-analyticity: Theorem 3.5 shows that a *-analytical inhomogeneity q allows an explicit solution of the Navier-Stokes differential equation (3.21a) for arbitrary commutative pseudorings, and that furthermore for a special set of these pseudorings also the divergence condition (3.21b) holds. But for many applications, the inhomogeneity q will not be *-analytical but a gradient, and in this case Theorem 3.5 is applicable only on *-analytical gradients, which represents a strong restriction.Similar considerations hold for the Einstein equations: In Theorem3.3  *-analyticity appears in the stress energy tensor (in a hidden form) as well as in the metric tensor, but a lot of physical interesting stress energy tensors will not be *-analytical.These considerations boil down to the following question: is it possible to embed an arbitraryn-dimensional smooth vector field f (tensor field) into an N-dimensional *-analytical vector field F (tensor field) for a sufficient large dimension N≥n in the sense, that the first n components of F allow a representation of the n components of f? (An exact definition of “embedding” will be given in Section 4.)In the case of a positive answer,we can embed an-dimensional nonlinear partial differential equation defined by only smooth vector fields onto aN-dimensional nonlinear partial differential equation defined by*-analytical vector fields, and this high-dimensional differential equation can be solved in the framework of GAP-Theory as demonstrated in the sections above.Finally the firstn components of the obtained N-dimensional solution vector field (tensor field) represent the solution of the n-dimensional start problem.Section4 will show that such an Embedding formalism of smoothness into *-analyticity (or more general: chain analyticity) exists.


## 4. Embedding Problems for GAPs. New Products

Embedding problems (also denoted as Immersion problems) are important in mathematical physics as well as in nonlinear system theory [14, 16, 26, 27] and can be formulated in various ways. Roughly spoken immersion theory asks if an n-dimensional structure with small symmetry can be seen as a “projection” of an N-dimensional structure with high symmetry, n≤N. Of course the answer depends on the concretely given symmetry classes as well as on the dimensions n and N.In the following, we are interested on a special class of embedding  problems, defined by the question if smoothness (which is a very  general symmetry) can be embedded into chain-analyticity (which  is a rather special symmetry).Definition 4.1.
Given ann-dimensional smooth vector field f∈𝔗1(Wn), Wn⊆ℝn, and an N-dimensional chain-analytic vector field F∈𝔗1(WN;*1,…,*M)(z0), 𝔛=(ℝN;*1,…,*M) an arbitrary PA1-chain, then f is denoted as embaddable intoF if it holds that
(4.1)fγ(z)=Fγ(Z)∣Zn+1=Zn+2⋯=ZN=0,withZγ=zγ,γ∈I1,n.
If this relation holds for arbitrary f∈𝔗1(Wn), then 𝔗1(Wn) is denoted as embaddable into 𝔗1(WN;*1,…,*M)(z0).Of course the possibility of embedding𝔗1(Wn) into 𝔗1(WN;*1,…,*M)(z0) will depend on the concrete form of the given PA1-chain 𝔛 as well as on the proportion of n to N. Is 𝔗1(Wn) embaddable for N-dimensional Kronecker PA1-chains, N finite and/or for N-dimensional Splitting PA1-chains, N finite? Or must N be infinite? We will answer these questions by specializing on the subset 𝔗[P]1(Wn)⊆𝔗1(Wn) defined by all polynomials of order p≤P∈ℕ, and by specializing PA1-chains to pseudorings.Theorem 4.2.
Given the following classes ofN-dimensional pseudorings  𝔗1(ℝN;*)(z0):
(4.2)C(*)ijk(μ)=δHN(i+HN(uμ(j)))k,uμ(j):=jμ,μ>log2(n+1),μ∈ℕ,i,j,k∈I1,N,
with the index function  HN from (2.55), then the following statements hold. Statement 1.
For every given pairn,P∈ℕthere exist finite numbers  N(n,P,μ)∈ℕsuch that 𝔗[P]1(Wn)is embaddable into 𝔗[P]1(WN;*)(0),0∈Wn⊂WN.Statement 2.
A sufficient condition for embedding is given by(4.3)N≥Nmin(n,P,μ),withNmin(n,P,μ):=n+nμP.Statement 3.
Leta[p]β1β2⋯βpγ with  γ,β1,…,βp∈I1,n,p∈I0,P denote the coefficients of an arbitrary element f∈𝔗[P]1(Wn).Then under the assumption (4.3) the coefficients  A[p]i,p∈I0,P,  i∈I1,Nof the corresponding element F∈𝔗[P]1(WN;*)(0)are defined by
(4.4)A[0]γ=a[0]γ,A[p]γ+∑k=1p(βk)μ‍=a[p]β1β2⋯βpγ,p∈I1,P.Proof
Step 1.
The pseudoring symmetry of𝔗1(ℝN;*)(z0): for a given μ, it is
(4.5)C(*)ijl(μ)C(*)lkm(μ)=δHN(i+HN(u(j)))lδHN(l+HN(u(k)))m=δHN(HN(i+HN(u(j)))+HN(u(k)))m=δHN(i+HN(u(j))+u(k))m=δHN(i+u(j)+u(k))m,⇒C(*)i[jl(μ)C(*)|l|k]m(μ)=0,
where we have used the property HN(HN(k)+m)=HN(k+m), and so forth.Step 2.
The conditions of embaddability𝔗[P]1(Wn)into𝔗[P]1(WN;*1,…,*P)(0)in general PA1-chains: By definition every element f∈𝔗[P]1(Wn),0∈Wn has a representation (In the following we use Greek indices for the range I1,n, Latin indices for the ranges I1,N and I0,P.)
(4.6)fγ(z)=a[0]γ+a[1]β1γzβ1+a[2]β1β2γzβ1zβ2+⋯+a[P]β1β2⋯βPγzβ1zβ2⋯zβP,z∈Wn, where the coefficients a[p]β1⋯βpγ∈ℝ for  γ,β1,…,βp∈I1,n,  p∈I0,P can be assumed as total symmetric in the indices β1,…,βp. Furthermore, for a general given PA1-chain 𝔛=(ℝN;*1,…,*P) every element F∈𝔗[P]1(WN;*1,…,*P)(0) has a representation
(4.7)F(Z)=A[0]+A[1]*1Z+(A[2]*2Z)*1Z+((A[3]*3Z)*2Z)*1Z+⋯+(⋯((A[M]*PZ)*P-1Z)*P-2⋯)*1Z,Z∈WN
with certain vector-valued coefficients A[p]∈ℝN,  p∈I0,P. In component representation,
(4.8)Fk(X)=A[0]k+A[1]iC(*1)ijkZj+A[2]iC(*2)ij1l1Zj1C(*1)l1j2kZj2+⋯+A[P]iC(*P)ij1l1Zj1C(*P-1)l1j2l2Zj2⋯C(*1)lP-1jPkZjP=rewrittenA[0]k+A[1]iC(*1)ijkZj+A[2]iC(*2)ij1l1C(*1)l1j2kZj1Zj2+⋯+A[P]iC(*P)ij1l1C(*P-1)l1j2l2⋯C(*1)lP-1jPkZj1Zj2jP.
The restriction on the subset Wn⊆WN by Zn+1=Zn+2⋯=ZN=0,Zα=zα,α∈I1,n leads to
(4.9)Fγ(Z)|Zn+1=Zn+2⋯=ZN=0=A[0]γ+A[1]iC(*1)iβ1γzβ1+A[2]iC(*2)iβ1l1C(*1)l1β2γzβ1zβ2+⋯+A[P]iC(*P)iβ1l1C(*P-1)l1β2l2⋯C(*1)lP-1βPγzβ1zβ2⋯zβP,γ∈I1,n,
and the embedding condition (4.1) takes the form
(4.10)a[0]γ+a[1]β1γzβ1+a[2]β1β2γzβ1zβ2+⋯+a[P]β1β2⋯βPγzβ1zβ2⋯zβP=A[0]γ+A[1]iC(*1)iβ1γzβ1+A[2]iC(*2)iβ1l1C(*1)l1β2γzβ1zβ2+⋯+A[P]iC(*P)iβ1l1C(*P-1)l1β2l2C(*1)lP-1βPγzβ1zβ2⋯zβP,γ∈I1,n.
Comparison of the components leads to the following equation system:
(4.11)A[0]γ=a[0]γ,A[p]iC(*p)iβ1l1C(*p-1)l1β2l2⋯C(*1)lp-1βpγ=a[p]β1β2⋯βpγ,(b),γ,β1,…,βp∈I1,n,  p∈I1,P,  or in detailed denotion:
(4.12)A[0]γ=a[0]γ,A[1]iC(*1)iβ1γ=a[1]β1γ,A[2]iC(*2)iβ1l1C(*1)l1β2γ=a[2]β1β2γ,A[3]iC(*3)iβ1l1C(*2)l1β2l2C(*1)l2β3γ=a[3]β1β2β3γ,⋮A[P]iC(*P)iβ1l1C(*P-1)l1β2l2⋯C(*1)lP-1βPγ=a[P]β1β2⋯βPγ,γ,β1,…,βp∈I1,n. We see that the total symmetry in the covariant indices of the right side is compatible with the total symmetry in the covariant indices of the left side which follows immediately from the PA1-chain symmetry. The question is now if this system is solvable for arbitrary given n,P and components a[P]β1⋯βpγ,γ,β1,…,βp∈I1,n,p∈I0,P, that is, if we can find an adequate dimension N(n,P), adequate coefficients A[α]i,α∈I0,P,i∈I1,N, and an adequate PA1-chain with structure constants C(*α)ijk such that (4.12) get solvable. Since not only the coefficients A[α]i but also the structure constants are unknown, the equation system is nonlinear.Step 3.
The conditions of embaddability𝔗[P]1(Wn)into𝔗[P]1(WN;*)(0)in the special PA1-chain (4.2): now we have to insert the pseudoring structure (4.2) in the equation system above. For this we calculate
(4.13)C(*2)iβ1l1C(*1)l1β2γ=δHN(i+HN(uμ(β1)))l1δHN(l1+HN(uμ(β2)))γ=δHN(HN(i+HN(uμ(β1)))+HN(uμ(β2)))γ=see step 1 aboveδHN(i+uμ(β1)+uμ(β2))γ,C(*3)iβ1l1C(*2)l1β2l2C(*1)l2β3γ=δHN(i+HN(uμ(β1)))l1δHN(l1+uμ(β2)+uμ(β3))γ=δHN(HN(i+HN(uμ(β1)))+uμ(β2)+uμ(β3))γ=see step 1 aboveδHN(i+uμ(β1)+uμ(β2)+uμ(β3))γ,
and we obtain for the general case
(4.14)C(*p)iβ1l1C(*p-1)l1β2l2⋯C(*1)lp-1βpγ=δHN(i+uμ(β1)+uμ(β2)+⋯+uμ(βp))γ=insertinguμ(βm)δHN(i-(β1)μ-(β2)μ-⋯-(βp)μ)γ=δHN(i-∑k=1p(βk)μ)γ,p∈I1,P.
Inserting in the equation system (4.11), we get(4.15a)A[0]γ=a[0]γ,(4.15b)A[p]HN(γ+∑k=1p(βk)μ)=a[p]β1β2⋯βpγ,γ,β1,β2,…,βp∈I1,n,  p∈I1,P. Now we have to show that for arbitrary given n,P and arbitrary components a[P]β1⋯βpγ, there exist natural numbers μ(n,P) and N(n,P,μ) and coefficients A[α]i,  α∈I0,P,  i∈I1,N with the property above. The positive answer will be given in the next step.Step 4.
The solvability of the equation system (4.15a) and (4.15b): for the following considerations, we define the set
(4.16)I̅1,n[p+1]:={(γ,β1,β2,…,βp)withγ,βi∈I1,n,βi+1≥βi},p∈I1,P,
that is, I̅1,n[p+1] is a subset of the set I1,n[p+1]:=I1,n×I1,n×⋯×I1,n,  (p+1 times) defined by the additional property βi+1≥βi. We see that I̅1,n[p+1] is the definition region of the index field (γ,β1,β2,…,βp) from the coefficients a[p]β1β2⋯βpγ in (4.15b),  since for these coefficients the total symmetry in the covariant indices hold by assumption. For the following, it is advantageous to write β instead of (β1,β2,…,βp) and (γ,β) instead of (γ,β1,β2,…,βp). With these denotions, we can formulate a necessary and sufficient condition for the solvability of (4.15a) and (4.15b) for arbitrary given n,P, and for arbitrary given coefficients a[p]β1β2⋯βpγ as follows:
(4.17)HN(γ+∑k=1p(βk)μ)≠HN(γ̅+∑k=1p(β̅k)μ‍)for(γ,β)≠(γ̅,β̅),
with (γ,β),(γ̅,β̅)∈I̅1,n[p+1],p∈I1,P (if the following relation is not satisfied, that is, if there exists for a fixed p at least one pair of (p+1)-tuples (γ,β)≠(γ̅,β̅) such that HN(γ+∑k=1p(βk)μ)=HN(γ̅+∑k=1p(β̅k)μ), the condition (4.15b) would imply for the fixed p: a[p]β1β2⋯βpγ=a[p]β̅1β̅2⋯β̅pγ̅,  which means a specialization by the coefficients a[p]β1β2⋯βpγ). We have to show that this relation holds under our assumptions of μ(n,P) and N(n,P,μ)≥Nmin(n,P,μ) as stated in (4.2), (4.3). In the case N(n,P,μ)≥Nmin(n,P,μ)=n+nμP, it is γ+∑k=1p(βk)μ≤Nmin(n,P,μ)  for p∈I1,P  which implies HN(γ+∑k=1p(βk)μ‍)=γ+∑k=1p(βk)μ‍ for p∈I1,P. Inserting in (4.15a) and (4.15b), we obtain the statement (4.4). The solvability condition (4.17) takes the form γ+∑k=1p(βk)μ‍≠γ̅+∑k=1p(β̅k)μ‍ or rewritten as
(4.18)∑k=1p(βk)μ‍-∑k=1p(βk)μ‍≠γ̅-γ,for(γ,β)≠(γ̅,β̅),(γ,β),(γ̅,β̅)∈I̅1,n[p+1],p∈I1,P. From the considerations above, we see that this condition is sufficient for the solvability of the the system (4.15b) for general given n,P,a[p]β1β2⋯βpγ. Since γ̅-γ moves in the region [-n,n], the condition is certainly satisfied by demanding
(4.19)|∑k=1p(β̅k)μ‍-∑k=1p(β̅k)μ‍|>n,forβ≠β̅,β,β̅∈I̅1,n[p+1],p∈I1,P. The worst case of this new condition is given if only one component of β̅ differs from one component of β. Let k0 denote the corresponding index, then we obtain from above the worst case in the form |(β̅k0)μ-(β̅k0)μ|>n,  for βk0≠β̅k0. The worst case of this scenario is given by the minimal possible difference, that is, by βk0=1,β̅k0=2. In this case μ is defined by the sufficient condition |1-2μ|>n or 2μ>n+1  which is satisfied by μ>log2(n+1) as stated in (4.2).I will denote this Theorem as thePseudoring Embedding Theorem, and the pseudoring (4.2) as embedding pseudoring. The Theorem shows that a non *-analytic vector field always can be embedded into a higher-dimensional *-analytic vector field. Furthermore, this Theorem not only characterizes the possibility of embedding but also allows an explicit description of the practical embedding process. Some remarks for a deeper understanding are as follows.(i)
Since the mapuμ in the embedding pseudoring is not a linear map, the embedding pseudoring is neither commutative nor associative: embedding is not compatible with commutativity and associativity.(ii)
The embedding pseudoring is a Kronecker algebra, which leads to the question if also splitting pseudorings can be used for embedding. It can be shown that this is not possible in a general framework, that is, for embedding problems Kronecker algebras are more important than splitting algebras. Otherwise, the applications in   [17] show that in the framework of solving partial differential equations the splitting algebras seem to be more important than Kronecker algebras. The upshot is that both algebra-types are important.(iii)
The embedding condition (4.3) on the embedding dimension N≥Nmin is a sufficient condition. It is possible that there exist cases where a smaller dimension N≤Nmin is purposeful.(iv)
Until now our embedding analyse was restricted on polynomials. It can be generalized on arbitrary smooth vector fields, but in this case the embedding dimensionN will not remain finite, according to the embedding relation (4.3). Thus it would be interesting to investigate if (4.3) is not only sufficient but also necessary.(v)
Relation (4.4) shows that the coefficients A[p]i of the vector field F are not defined entirely by the coefficients a[p]β1β2⋯βpγ of the vector field f: a lot of coefficients A[p]i remain unrestricted. In other words, embedding from f into F does not define F entirely but only partially.(vi)
Until now our investigation was fixed on the case of vector-field embedding problems. More general is the case of tensor field embedding problems, which can be solved in an analogue way. An analyse is omitted here for space reason.As a simple example, we consider the case of embedding an(n=2)-dimensional polynomial of order P=3. Then the parameter μ∈ℕ is restricted by the condition μ>log2(n+1), and the smallest natural solution is given by μ=2. The embedding dimension N is restricted by the condition N≥Nmin(n,P,μ)=n+nμP=2+22·3=14, and we take the smallest value N=14. According to (4.4), the coefficients A[p]i of the embedding polynomial F take the form
(4.20)p=0:γ=1:A[0]1=a[0]1,γ=2:A[0]2=a[0]2,p=1:γ=1:A[1]2=a[1]11,A[1]5=a[1]21,γ=2:A[1]3=a[1]12,A[1]6=a[1]22,p=2:γ=1:A[2]3=a[2]111,A[2]6=a[2]121,A[2]9=a[2]221,γ=2:A[2]4=a[2]112,A[2]7=a[2]122,A[2]10=a[2]222,p=3:γ=1:A[3]4=a[3]1111,A[3]7=a[3]1121,A[3]10=a[3]1221,A[3]13=a[3]2221,γ=2:A[3]5=a[3]1112,A[3]8=a[3]1122,A[3]11=a[3]1222,A[3]14=a[3]2222,
and according to the remarks above the remaining coefficients A[p]i can be chosen to zero, that is,  A[0]3=A[0]4=⋯=A[0]14=0,A[1]1=A[1]4=A[1]7=A[1]8=⋯=A[0]14=0, and so forth.Finally we will discuss a certain extension of our pseudoring-theory, which is interesting for certain embedding problems. We remember that until now we only have dealed with products⊛(*):Vp×V→Vp and ⊛(*)dual:Vq×V→Vq, which leads to the question if “similar” products Vp×Vp→Vp and Vq×Vq→Vq can be defined with similar properties. This will be done now.Definition 4.3.
Given an arbitrary algebra(V;*) with structure constant tensor C(*), then the*-associated product of order p:⊠(*):Vp×Vp→Vp is defined by
(4.21)(X⊠(*)Y)k1k2⋯kp:=Xi1i2⋯ipC(*)i1j1k1C(*)i2j2k2⋯C(*)ipjpkpYj1j2⋯jp,
and the dual-associated product of order q:⊠(*)dual:Vq×Vq→Vq is defined by
(4.22)(X⊠(*)dualY)k1k2⋯kq:=Xi1i2⋯iqC(*)k1j1i1C(*)k2j2i2⋯C(*)kqjqiqYj1j2⋯jq.Proposition 4.4.
Given a PA1-structure(V;*1,*2), then the following symmetry holds:
(4.23)(X⊠(*1)Y)⊠(*2)Z=(X⊠(*1)Z)⊠(*2)Y,∀X,Y,Z∈Vp.Proof.
It is(4.24){(X⊠(*1)Y)⊠(*2)Z}m1m2⋯mp=(X⊠(*1)Y)k1k2⋯kpC(*2)k1l1m1C(*2)k2l2m2⋯C(*2)kplpmpZl1l2⋯lp=Xi1i2⋯ipC(*1)i1j1k1C(*1)i2j2k2⋯C(*1)ipjpkpYj1j2⋯jpC(*2)k1l1m1C(*2)k2l2m2⋯C(*2)kplpmpZl1l2⋯lp=rewrittenXi1i2⋯ipYj1j2⋯jpZl1l2⋯lpC(*1)i1j1k1C(*1)i2j2k2⋯C(*1)ipjpkpC(*2)k1l1m1C(*2)k2l2m2⋯C(*2)kplpmp.
Since C(*1)iα[lαkαC(*2)|kα|jα]mα=0,α∈I1,p, the term above can be written as
(4.25)Xi1i2⋯ipYj1j2⋯jpZl1l2⋯lpC(*1)i1l1k1C(*1)i2l2k2⋯C(*1)iplpkpC(*2)k1j1m1C(*2)k2j2m2⋯C(*2)kpjpmp=rewrittenXi1i2⋯ipC(*1)i1l1k1C(*1)i2l2k2⋯C(*1)iplpkpZl1l2⋯lpC(*2)k1j1m1C(*2)k2j2m2⋯C(*2)kpjpmpYj1j2⋯jp=rewritten(X⊠(*1)Z)k1k2⋯kpC(*2)k1j1m1C(*2)k2j2m2⋯C(*2)kpjpmpYj1j2⋯jp={(X⊠(*1)Z)⊠(*2)Y}m1m2⋯mp.In other words, the PA1-symmetry is induced from the operations*1,*2 to the operations ⊠(*1),⊠(*2). The next question is if the property of *-analyticity is preserved, that is, if X,Y∈𝔗2(W;*)(ξ0) induces (X⊠(*)Y)∈𝔗2(W;*)(ξ0), and if X∈𝔗2(W;*)(ξ0) and Y∈𝔗2(W;*)(ξ0) induce (X⊠(*)dualY)∈𝔗2(W;*)(ξ0). The next proposition gives a positive answer.Proposition 4.5.
For a pseudoring(ℝn;*), the following symmetries hold:
(4.26)X,Y∈𝔗2(W;*)(ξ0)⇒(X⊠(*)Y)∈𝔗2(W;*)(ξ0)if(ℝn;*)is commutative  or the vector fieldYis a constant,(4.27)X∈𝔗2(W;*)(ξ0),Y∈𝔗2(W;*)(ξ0)⇒(X⊠(*)dualY)∈𝔗2(W;*)(ξ0)if(ℝn;*)is commutative  or vector fieldYis a constant.Proof.
To show (4.26) we have to show that for the product (X⊠(*)Y)the PCRE hold under the given assumptions, that is,
(4.28)(X⊠(*)Y),mkl=[𝔇(*)(X⊠(*)Y)]slC(*)smk+[𝔇(*)(X⊠(*)Y)]ksC(*)sml
with adequate  𝔇(*)(X⊠(*)Y). With Zkl:=(X⊠(*)Y)kl=XijC(*)ipkC(*)jqlYpq,  we calculate
(4.29)Z,mkl=X,mijC(*)ipkC(*)jqlYpq+XijC(*)ipkC(*)jqlY,mpq=PCRE[(𝔇(*)X)sjC(*)smi+(𝔇(*)X)isC(*)smj]C(*)ipkC(*)jqlYpq+XijC(*)ipkC(*)jql[(𝔇(*)Y)sqC(*)smp+(𝔇(*)Y)psC(*)smq].
In the case of a constant Y, we obtain
(4.30)Z,mkl=[(𝔇(*)X)sjC(*)smiC(*)ipkC(*)jql+(𝔇(*)X)isC(*)smjC(*)ipkC(*)jql]Ypq=pseudoringsymmetry[(𝔇(*)X)sjC(*)spiC(*)imkC(*)jql+(𝔇(*)X)isC(*)sqjC(*)ipkC(*)jml]Ypq=[(𝔇(*)Z)ilC(*)imk+(𝔇(*)Z)kiC(*)iml]Ypq,with(𝔇(*)Z)il:=(𝔇(*)X)sjC(*)spiC(*)jqlYpq,
and the second statement of (4.26) has been proved. To show the first statement, we start from the partial derivations above.
(4.31)Z,mkl=[(𝔇(*)X)sjC(*)smiC(*)ipkC(*)jql+(𝔇(*)X)isC(*)smjC(*)ipkC(*)jql]Ypq+Xij[(𝔇(*)Y)sqC(*)smpC(*)ipkC(*)jql+(𝔇(*)Y)psC(*)smqC(*)ipkC(*)jql]=comm. pseudoring[(𝔇(*)X)sjC(*)spiC(*)imkC(*)jql+(𝔇(*)X)isC(*)sqjC(*)ipkC(*)jml]Ypq+Xij[(𝔇(*)Y)sqC(*)sipC(*)mpkC(*)jql+(𝔇(*)Y)psC(*)sjqC(*)ipkC(*)mql]=renamingindices[(𝔇(*)X)ijC(*)ipsC(*)smkC(*)jql+(𝔇(*)X)ijC(*)jqsC(*)ipkC(*)sml]Ypq+Xij[(𝔇(*)Y)pqC(*)pisC(*)mskC(*)jql+(𝔇(*)Y)pqC(*)qjsC(*)ipkC(*)msl]=C(*)smk[(𝔇(*)X)ijC(*)ipsC(*)jqlYpq+Xij(𝔇(*)Y)pqC(*)pisC(*)jql]+C(*)sml[(𝔇(*)X)ijC(*)jqsC(*)ipkYpq+Xij(𝔇(*)Y)pqC(*)qjsC(*)ipk]=C(*)smk(𝔇(*)Z)sl+C(*)sml(𝔇(*)X)ks,with(𝔇(*)Z)sl:=(𝔇(*)X)ijC(*)ipsC(*)jqlYpq+Xij(𝔇(*)Y)pqC(*)pisC(*)jql=[(𝔇(*)X)⊠(*)Y+X⊠(*)(𝔇(*)Y)]sl.
The proof of the statement (4.27) runs the same lines.We see that the property of*-analyticity generally will not be preserved by the operations ⊠(*) and ⊠(*)dual, only for simple underlying structures (commutative pseudorings) or for a simple operand (a constant). Thus it makes sense to introduce the concept of weak*-analytical fields as ⊠(*)-products and ⊠(*)dual-products of *-analytical fields, which allows applications in general relativity as well as in system theory, but will not be discussed here for space reason.In this chapter, a class of Embedding Problems has been analyzed with   the result that polynomialn-dimensional vector fields always can   be embedded into polynomialN-dimensional,*-analytic vector fields,N≥nfinite. In the caseN→∞,the results hold for arbitrary smooth   vector fields.  Furthermore, not only the existence of embedding was  shown, but the embedding process was described explicitly. Finally   new product operations with interesting properties have been analyzed.


## 5. Conclusion and Outlook

The paper presents and solves a new type of Embedding Problems which are important for solving nonlinear partial differential equations. The most important results are as follows.(i)
A set of algebras was found (so-called “PA1-chains” or “GAPs”), such that for the corresponding function theories a generalization of the Cauchy-Riemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.(ii)
A wide variety of GAPs can be characterized explicitly by presenting their structure constant tensors explicitly.(iii)
GAP-Theory endows new explicit solutions of Einsteins field equations from General Relativity Theory and new explicit solutions of then-dimensional Navier-Stokes equations.(iv)
A new Embedding Theorem was formulated, which allows a wide generalization of the results above: it was shown explicitly that an arbitraryn-dimensional smooth vector field always can be embedded into a special N-Dimensional Smooth Vector Field (N≥n) restricted by PA1-symmetries. In other words, the world of PA1-chains is wide enough to allow embedding for rather general (“smooth”) structures.(v)
Some consequences of the Embedding Theorem for the explicit solution of nonlinear partial differential equations have been discussed.(vi)
A simple example for the Embedding Theorem has been given.The following investigations and generalizations would be useful, the construction of more general PA1-chains which would generalize our Embedding Theorem.



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*Source: 120213-2010-04-15.xml*