partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

smooth solutions of the euler and navier stokes equations from the a posteriori analysis of approximate solutions abstract the main result of morosi and pizzocchero 2012 is presented in a variant based on a c formulation of the cauchy problem in this approach the a posteriori analysis of an approximate solution gives a bound on the sobolev distance of any order between the exact and the approximate solution

implication structures fuzzy subsets and enriched categories we show that there are natural ways to make a fuzzy set into an enriched category and that in doing so some basic results in fuzzy set theory are applications of enriched category theory thus we argue that the theory of enriched categories is a useful tool for fuzzy set theorists

sheaves on involutive quantales we show that sheaves on grothendieck quantales which are the quantales that correspond to grothendieck toposes may be defined in several equivalent ways the main topic of this paper is the characterization of internal posets and suplattices in the topos of sheaves on a grothendieck quantale q in terms of categories enriched in a suitable quantaloid and by means of q modules we thereby extend the existing results concerning locales

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

a note on the hypergraph functor the purpose of this note is to show that in the case of spatial locales l the hypergraph functor has a right adjoint functor

category theoretic aspects of chain valued frames part ii applications to lattice valued topology this paper is part ii of a two part series dealing with category theoretic aspects of chain valued frames using the categorical properties established for l frm in part i for l a complete chain this paper constructs upper free functor l and lower free functor r the functor l is used to create a class of non generated l frames factor the lpt spectrum functor through the s spectrum functor resolve the relationship between l sobriety and i l sobriety and give insight into the construction of universal l topological spaces and the functor r is used to create a class of non generated l frames factor the w l functor through a new spectrum functor s and give insight into the construction of co universal l topological spaces these facts universal spaces are anti stratified co universal spaces are stratified l frm produces both kinds of spaces via l and r l frm via its dual l loc is adjunctive with l top comprise a coherent argument that l top must accommodate both kinds of spaces resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of l topologies

ambiguous representations as fuzzy relations between sets crisp and l fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced it is shown that for each pair of compact hausdorff spaces the set of crisp or l fuzzy ambiguous representations is a lattice and a compact hausdorff lawson upper semilattice the categories of ambiguous and l ambiguous representations are defined and investigated

conditioning and dichotomy in differential algebraic equations the relationship between the dichotomy of the solution space for a differential algebraic system and the well conditioning of associated boundary value problems is investigated this work extends results that are known for regular explicit ordinary differential equations and it follows the presentation by de hoog and mattheij siam j numer anal 24 1987 pp 89 105

fuzzy normal subgroups and fuzzy cosets abstract we introduce here the notion of 1 i a fuzzy normal subgroup and 2 ii a fuzzy coset we prove that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g further we show that the level subgroups of a fuzzy normal subgroup are all normal we obtain several results which are analogs of some basic theorems of group theory in particular a fuzzy version of lagrange s theorem is established

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

new answers to an old question in the theory of differential algebraic equations in the context of linear differential algebraic equations daes one finds different associated explicit ordinary differential equations odes among them essential underlying and inherent explicit regular ones abbreviated euodes and ierodes euodes have been introduced in 1991 for index 2 daes in hessenberg form by means of special transformations ierodes result within the framework of the projector based decoupling each such explicit ode is occasionally considered to rule the flow of the dae the question to which extend euodes and ierodes are related to each other has been asked promptly after 1991 for index 2 hessenberg form daes answers have been given in 2005 saying that euodes represent somehow condensed ierodes recently euodes have been indicated for general arbitrary index daes and it has been proved that they are condensed ierodes the understanding of the relation between the ierode and the euodes enables to uncover the stability behavior of the dae flow in the present paper we show that both the ierodes and euodes of a dae with arbitrary high index do not at all depend on derivatives of the right hand side we consider adjoint pairs of daes and provide generalizations of the classical lagrange identity furthermore we address lyapunov spectra and lyapunov regularity

solving linear equations by fuzzy quasigroups techniques abstract we deal with solutions of classical linear equations a x b and y a b applying a particular lattice valued fuzzy technique our framework is a structure with a binary operation a groupoid equipped with a fuzzy equality we call it a fuzzy quasigroup if the above equations have unique solutions with respect to the fuzzy equality we prove that a fuzzy quasigroup can equivalently be characterized as a structure whose quotients of cut substructures with respect to cuts of the fuzzy equality are classical quasigroups analyzing two approaches to quasigroups in a fuzzy framework we prove their equivalence in addition we prove that a fuzzy loop quasigroup with a unit element which is a fuzzy semigroup is a fuzzy group and vice versa finally using properties of these fuzzy quasigroups we give answers to existence of solutions of the mentioned linear equations with respect to a fuzzy equality and we describe solving procedures quasigroups and other related structures are an algebraic tool successfully applied up to now in coding theory and cryptology in our work we propose related applications

from quantale algebroids to topological spaces fixed and variable basis approaches using the category of quantale algebroids the paper considers a generalization of the classical papert papert isbell adjunction between the categories of topological spaces and locales to partial algebraic structures it also provides a single framework in which to treat the concepts of quasi standard and stratified fuzzy topology

fuzzy congruences in groups the purpose of this paper is to introduce some basic concepts of fuzzy congruence and to prove some fundamental properties we obtain several results which are analogs of some basic theorems of group theory in particular we prove that in a group g the lattice of some fuzzy normal subgroups is modular

fuzzy relations on rings and groups abstract let s be any nonempty set a fuzzy relation on s is a fuzzy subset of s s in this paper we study fuzzy relations on rings and groups in particular we show that if and are fuzzy left right ideals of a ring r then is a fuzzy left right ideal of r r and conversely if is a fuzzy left right ideal of r r then either or is a fuzzy left right ideal of r an example is given to show that if is a fuzzy left right ideal of r r then and both need not be fuzzy left right ideals of r an example is also given to show that if is a fuzzy left right ideal of r r then the weakest fuzzy subset of r on which is a fuzzy relation need not be a fuzzy left right ideal of r we obtain similar results for groups we also show that certain results of bhattacharya and mukherjee 1985 are not true

transfer of boundary conditions for daes of index 1 in this paper the concept of abramov s method for transferring boundary conditions posed for regular ordinary differential equations odes is applied to index 1 differential algebraic equations daes having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of abramov s method we consider boundary value problems for index 1 linear daes both with constant and varying leading matrices we describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval the index 1 daes which realize the transfer are given and their properties are studied the results are reformulated for inhomogeneous index 1 daes as well

fuzzy presubsets as non idempotent and non commutative classifications of subalgebras in the framework of quantale preordered sets regular q fuzzy presubsets are identified with classifiable subalgebras this identification is based on the monad t of quasi presingletons which is an extension of the monad associated with the cauchy completion of quantale preordered sets even though not every t subalgebra is classifiable the classifiable hull of t subalgebras exists always an application of these constructions is the internalization of the negation of the underlying quantale as truth arrow in the category of t algebras

on the stability of the abramov transfer for differential algebraic equations of index 1 the transfer of boundary conditions for ordinary differential equations developed by abramov zh vychisl mat mat fiz 1 1961 pp 542 545 is a stable method for representing the solution spaces of linear boundary value problems instead of boundary value problems matrix valued initial value problems are solved when integrating these differential equations the inner independence of the columns of the solution matrix and hence of the solutions of the resulting linear system of equations remains valid balla and marz have generalized abramov s transfer for homogenized index 1 differential algebraic equations siam j numer anal 33 1996 pp 2318 2332 in this article a direct version of the abramov transfer for inhomogeneous linear mbox index 1 differen linebreak 4 tial algebraic equations is developed and the numerical stability of this method is proved

fuzzy sets and geometric logic hohle has identified fuzzy sets valued in a frame complete heyting algebra w with certain sheaves over w the subsheaves of constant sheaves more general sheaves can be got as quotients of the fuzzy sets his principal approach to sheaves over w and topos theoretic constructions on them is via complete w valued sets in this paper we show how the geometric fragment of those constructions can be described in a natural stalkwise manner provided one works also with incomplete w valued sets our exposition examines in detail the interactions between different technical expressions of the notion of sheaf and highlights a conceptual view of sheaf as continuous set valued map

maximal fuzzy maps fuzzy maps are studied by stressing the idea that they are respectful with granularity when it is modeled by the existence of fuzzy equivalence relations indistinguishability operators the granules of the system are then identified with their fuzzy points some properties of perfect and classical fuzzy maps are stated and the existence of maximal fuzzy maps is proved maximal fuzzy maps handle the greatest uncertainty in the sense that the granules are of the greatest possible size so they are the less specific maps in the system since granularity is expressed with fuzzy points they are also studied in the paper the existence of maximal fuzzy points is established and they are characterized as the fixed points of a special map that we call l e

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

a comprehensive theory of trichotomous evaluative linguistic expressions in this paper a logical theory of the so called trichotomous evaluative linguistic expressions tev expressions is presented these are frequent expressions of natural language such as small very small roughly medium extremely big etc the theory is developed using the formal system of higher order fuzzy logic namely the fuzzy type theory generalization of classical type theory first we discuss informally what are properties of the meaning of tev expressions then we construct step by step axioms of a formal logical theory t e v of tev expressions and prove various properties of t e v all the proofs are syntactical and so our theory is very general we also outline construction of a canonical model of t e v the main elegancy of our theory consists in the fact that semantics of all kinds of evaluative expressions is modeled in a unified way we also prove theorems demonstrating that essential properties of the vagueness phenomenon can be captured within our theory

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

fuzzy normal subgroups and fuzzy quotients abstract in case of groups a mormal subgroup n of a group g can be defined with three equivalent approaches namely i as a kernel of some group homomorphism ii as a subgroup commuting with every element of g and iii as the subgroup n closed with respect to conjugates of elements of n in case of generalizations of this concept to fuzzy sets the elementwise third approach is found more suitable and hence is widely adopted by many authors in this paper we further study this concept of fuzzy subgroups and introduce fuzzy quotients called fuzzy quotient groups for all 0 1 two different definitions of these fuzzy quotients have been given which are shown to lead to isomorphic structures we also study the direct products of fuzzy normal subgroups min product as per the terminology of sherwood 1983 and the problem of writing a fuzzy normal subgroups of a direct product of some groups as a direct product of certain fuzzy normal subgroups has been discussed in detail

fuzzy sets and sheaves part ii sheaf theoretic foundations of fuzzy set theory with applications to algebra and topology this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

product structure of the fuzzy factor groups abstract in this paper first we prove that the fuzzy factor group a b has the sup property if a has then we study the product structure of the fuzzy factor groups

linear subspaces for linear daes of index 1 abstract in this paper the representation of linear subspaces of solutions of index 1 daes a x y b x y 0 is given it is shown that the solutions of adjoint systems are strongly related to the subspaces

reduction of fuzzy strict order relations abstract fuzzy strict order relations and the notion of their reduction are defined a necessary and sufficient condition is obtained for the transitive closure of the reduction to coincide with the strict order itself some possible graph theoretic significances of these results are discussed in the conclusion

preordered sets valued in a gl monoid let m l be a gl monoid an m valued preordered set is an l subset endowed with a reflexive and m transitive l relation it is essentially a category enriched in a quantaloid generated by m this paper presents a study of m valued preordered sets with emphasis on symmetrization and the cauchy completion the main result states that symmetrization and the cauchy completion of m valued preordered sets commute up to a natural isomorphism

a natural interpretation of fuzzy mappings in this paper we present a new and natural interpretation of fuzzy mappings which is a direct extension of the natural interpretation of fuzzy sets and fuzzy relations shown in our previous paper a natural interpretation of fuzzy sets and fuzzy relations fuzzy sets and systems 128 2002 135 we interpret fuzzy mappings as mappings in a heyting valued model for intuitionistic set theory and present a characterization of fuzzy mappings with membership functions which seems to be different from all kinds of known definitions this interpretation is quite natural for it deduces naturally various properties of fuzzy mappings and it makes clear the meaning of the famous extension principle by zadeh which is said to be very important in fuzzy theory

decaying turbulence in the generalised burgers equation we consider the generalised burgers equation frac partial u partial t f u frac partial u partial x nu frac partial 2 u partial x 2 0 t geqq 0 x in s 1 where f is strongly convex and is small and positive we obtain sharp estimates for sobolev norms of u upper and lower bounds differ only by a multiplicative constant then we obtain sharp estimates for the dissipation length scale and the small scale quantities which characterise the decaying burgers turbulence i e the structure functions and the energy spectrum the proof uses a quantitative version of an argument by aurell et al j fluid mech 238 467 486 1992 note that we are dealing with decaying as opposed to stationary turbulence thus our estimates are not uniform in time however they hold on a time interval t1 t2 where t1 and t2 depend only on f and the initial condition and do not depend on the viscosity these results allow us to obtain a rigorous theory of the one dimensional burgers turbulence in the spirit of kolmogorov s 1941 theory in particular we obtain two results which hold in the inertial range on one hand we explain the bifractal behaviour of the moments of increments or structure functions on the other hand we obtain an energy spectrum of the form k 2 these results remain valid in the inviscid limit

a natural interpretation of fuzzy sets and fuzzy relations we present a new and natural interpretation of fuzzy sets and fuzzy relations where the basic notions and operations have quite natural meanings we interpret fuzzy sets and fuzzy relations in a cumulative heyting valued model for intuitionistic set theory and define the basic notions and operations naturally in the model as far as fuzzy sets and fuzzy relations are considered as extensions of crisp sets and relations this interpretation seems to be most natural in the interpretation the canonical embedding from the class of all sets into the model plays an important role we distinguish generalized fuzzy sets fuzzy subsets of crisp sets and membership functions of fuzzy sets on crisp sets thus we present a foundation for developing a general theory of fuzzy sets where fuzzy subsets of different sets can be treated in a natural and uniform way

e fuzzy groups an e fuzzy group is a lattice valued algebraic structure defined on a crisp algebra which is not necessarily a group the crisp equality is replaced by a particular fuzzy one denoted by e the classical group like properties are formulated as appropriate fuzzy identities special lattice theoretic formulas we prove basic features of e fuzzy groups properties of the unit and inverses cancellability solvability of equations subgroup properties and others we also prove that for every cut of an e fuzzy group which is a classical subalgebra of the underlying algebra the quotient structure over the corresponding cut of the fuzzy equality is a classical group

eq algebras we introduce a new class of algebras called eq algebras an eq algebra has three basic binary operations meet multiplication and a fuzzy equality and a top element these algebras are intended to become algebras of truth values for a higher order fuzzy logic a fuzzy type theory ftt the motivation stems from the fact that until now the truth values in ftt were assumed to form either an imtl bl or mv algebra all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation multiplication and its residuum the latter is a natural interpretation of implication in fuzzy logic the equivalence is then interpreted by the biresiduum a derived operation the basic connective in ftt however is a fuzzy equality and therefore it is not natural to interpret it by a derived operation this defect is expected to be removed by the class of eq algebras introduced and studied in this paper from the algebraic point of view the class of eq algebras generalizes in a certain sense the class of residuated lattices and so they may become an interesting class of algebraic structures as such

fuzzy topologies and topological space objects in a topos abstract the purpose of this paper is to point out the link between lowen s fuzzy topologies and topological space objects in the topos l set as the main result we obtain the following theorem lowen s fuzzy topologies are the external version of the internal topologies in l set

category theoretic aspects of chain valued frames part i categorical and presheaf theoretic foundations this paper is part i of a two part series dealing with category theoretic aspects of chain valued frames a new categorical motivation for lattice valued frames is given from presheaves and then under the assumption that l be a complete chain it is established that standard category theoretic properties completeness cocompleteness factorization structures hold for l frm the category of l frames and l frame morphisms properties which are a platform for the upper free functor l and lower free functor r used in part ii to give a broad range of applications of chain valued frames to lattice valued topology

axiomatic of lattice valued f transform abstract axioms for upper and lower fuzzy transformation systems are introduced and it is proved that a transformation function satisfies axioms for upper or lower respectively transformation systems if and only if it is an upper or lower respectively f transform categories of upper lower respectively transformation systems are introduced and it is proved that both categories are isomorphic to the category of spaces with fuzzy partitions we prove that lower and upper f transforms preserve extensional fuzzy sets associated with spaces with fuzzy partitions

on approximate solutions of the equations of incompressible magnetohydrodynamics abstract inspired by an approach proposed previously for the incompressible navier stokes ns equations we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics mhd on a torus of arbitrary dimension d this setting involves a sobolev space of infinite order made of c vector fields with vanishing divergence and mean on the torus given any approximate solution of the mhd cauchy problem its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution and to bound from above the sobolev distance of any order between the exact and the approximate solution in certain cases the above mentioned lower bound on the time of existence is found to be infinite so one infers the global existence of the exact mhd solution we present some applications of this general scheme the most sophisticated one lives in dimension d 3 with the abc flow perturbed magnetically as an initial datum and uses for the cauchy problem a galerkin approximate solution in 124 fourier modes we illustrate the conclusions arising in this case from the a posteriori analysis of the galerkin approximant these include the derivation of global existence of the exact mhd solution with the abc datum when the dimensionless viscosity and resistivity are equal and stay above a critical value

the order theoretic duality and relations between partial metrics and local equalities for a gl monoid l hohle introduced l valued sets to formalize the mathematical theory of identity and existence in monoidal logic for the purpose of analyzing deadlock situations in data flow networks the notion of partial metric space was proposed by matthews later on a lattice theoretic generalization of pseudo partial metric spaces extending the range of pseudo partial metrics to a value lattice v were studied under the name of v pseudo pmetric spaces referring to a gl monoid l and a dual gl monoid v we show that l valued sets are order theoretically dual to v pseudopmetric spaces and apply this duality to the representation of v pseudopmetric spaces and the determination of their various categorical properties in addition to this the present paper provides not only some new results about the representation of l valued sets and their categories but also some non trivial categorical connections between l valued sets and v pseudopmetric spaces

estimates for solutions of a low viscosity kick forced generalized burgers equation we consider a non homogeneous generalised burgers equation u t f u u x nu 2u x 2 t r x s 1 here nu is small and positive f is strongly convex and satisfies a growth assumption while is a space smooth random kicked forcing term for any solution u of this equation we consider the quasi stationary regime corresponding to t 2 after taking the ensemble average we obtain upper estimates as well as time averaged lower estimates for a class of sobolev norms of u these estimates are of the form c nu with the same values of for bounds from above and from below they depend on and f but do not depend on the time t or the initial condition

membership functions for a fuzzy group from similarity relations abstract we show that the membership function x of a fuzzy group g satisfies the equation x e x where x y is a similarity relation on g which is invariant under left translation we also show that under some natural assumptions one can represent the elements x g as 1 1 onto transformations px on a suitable universe such that x equals the proportion of points in which are fixed points of px these results provide a deeper insight on the impact of group operation on the membership values x

generalized fuzzy topology versus non commutative topology the paper introduces a modification of the notions of generalized fuzzy topological space of demirci and quantal space of mulvey and pelletier suitable to explore interrelations between point set lattice theoretic topology and non commutative topology developed in the framework of c algebras or more recently of quantales as a consequence of the new approach a generalization of the concept of topological system of vickers arises moreover the currently dominating variable basis topological setting in the fuzzy community due to rodabaugh appears to be fixed basis

on ordered categories as a framework for fuzzification of algebraic and topological structures using the framework of ordered categories the paper considers a generalization of the fuzzification machinery of algebraic structures introduced by rosenfeld as well as provides a new approach to fuzzification of topological structures which amounts to fuzzifying the underlying set of a structure in a suitably compatible way leaving the structure itself crisp the latter machinery allows the so called double fuzzification i e a fuzzification of something that is already fuzzified

probability distribution functions for the random forced burgers equation statistical properties of solutions of the random forced burgers equation have been a subject of intensive studies recently see refs 1 6 of particular interest are the asymptotic properties of probability distribution functions associated with velocity gradients and velocity increments aside from the fact that such issues are of direct interest to a large number of problems such as the growth of random surfaces 1 it is also hoped that the field theoretic techniques developed for the burgers equation will eventually be useful for understanding more complex phenomena such as turbulence in this paper we propose a new and direct approach for analyzing the scaling properties of the various distribution functions for the random forced burgers equation we will consider the problem

topological aspects of non convergent sequences a comment on burgin s concept of fuzzy limits abstract it is well known that metric spaces can be provided with the structure of 0 1 valued sets on this background this paper explains an application of topologies on 0 1 valued sets to metric spaces and shows that every bounded sequence has always local limit points represented by singletons of the underlying 0 1 valued set

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

on approximate solutions of the incompressible euler and navier stokes equations abstract we consider the incompressible euler or navier stokes ns equations on a torus t d in the functional setting of the sobolev spaces h 0 n t d of divergence free zero mean vector fields on t d for n d 2 1 we present a general theory of approximate solutions for the euler ns cauchy problem this allows to infer a lower bound t c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u a and also to construct a function n such that u t u a t n n t for all t 0 t c both t c and n are determined solving suitable control inequalities depending on the error of u a the fully quantitative implementation of this scheme depends on some previous estimates of ours on the euler ns quadratic nonlinearity morosi and pizzocchero 2010 in press 7 8 to keep in touch with the existing literature on the subject our results are compared with a setting for approximate euler ns solutions proposed in chernyshenko et al 2007 1 as a first application of the present framework we consider the galerkin approximate solutions of the euler ns cauchy problem with a specific initial datum considered in behr et al 2001 9 in this case our methods allow amongst else to prove global existence for the ns cauchy problem when the viscosity is above an explicitly given bound

multidimensional potential burgers turbulence we consider the multidimensional generalised stochastic burgers equation in the space periodic setting frac partial mathbf u partial t nabla f mathbf u cdot nabla mathbf u nu delta mathbf u nabla eta quad t geq 0 mathbf x in mathbb t d mathbb r mathbb z d under the assumption that u is a gradient here f is strongly convex and satisfies a growth condition is small and positive while is a random forcing term smooth in space and white in time for solutions u of this equation we study sobolev norms of u averaged in time and in ensemble each of these norms behaves as a given negative power of these results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence namely the averages of the increments and of the energy spectrum these quantities behave as a power of the norm of the relevant parameter which is respectively the separation l in the physical space and the wavenumber k in the fourier space our bounds do not depend on the initial condition and hold uniformly in nu we generalise the results obtained for the one dimensional case in 10 confirming the physical predictions in 4 30 note that the form of the estimates does not depend on the dimension the powers of nu mathbf k ell are the same in the one and the multi dimensional setting

ω groups in the language of ω groupoids abstract we introduce groups as particular groupoids a structure with a single binary operation and an equality replacing the classical one the membership values belong to a complete lattice we analyze and compare languages in which such structures can be introduced we prove the equivalence of approaches to groups as algebras with three operations and those in the language of groupoids we also introduce a wider class of groups so called weak groups for which different neutral elements and different inverses of the same member are equal up to the equality for all these quotient structures with respect to cuts of the equality are classical groups we present basic features of groups in the language with one binary operation as an application we show that linear equations can be uniquely up to equality solved in these structures

fuzzy congruence relations abstract if a x f is an universal algebra and if r x denotes the set of all fuzzy subsets of x x i e fuzzy relations on x then the algebraic operations fg f on x are extended to algebraic operations f on r x f is an algebra similar to a it is called relational algebra a fuzzy equivalence relation which is at the same time a fuzzy subalgebra of r x f is called a fuzzy congruence relation on x in this paper we study fuzzy congruence relations amongst other things we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system

necessity of non stratified and anti stratified spaces in lattice valued topology this paper surveys the necessity of non stratified spaces from these viewpoints the characteristic functor the l spectrum and l soberification functors the upper free functor associated with l valued frames and two functorial embeddings associated with topological systems from semantic domains interestingly additional arguments also emerge en route for stratified spaces so that this paper ultimately argues for the necessity of both stratified and non stratified spaces in fixed basis topology the schemum of l top s and variable basis topology e g loc top

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

extension principles for closure operators on fuzzy sets and cuts abstract we investigate closure operators defined on various partly ordered sets of fuzzy objects including sets of all fuzzy sets with values in a complete residuated lattice and sets of all cuts systems defined in crisp sets and sets with similarity relations respectively we proved several extension theorems under which a closure operator defined on one universe can be extended to a closure operator defined on another universe we also investigate relationships between continuity of pairs of maps with respect to different closure operators

generating fuzzy factor groups and the fundamental theorem of isomorphism abstract in this paper we study how a fuzzy set can generate a fuzzy factor group we also study the product structure of the generating fuzzy factor group at the end of this paper we prove the first fundamental theorem of isomorphisms of fuzzy groups

geodesics in two dimensional first passage percolation we consider standard first passage percolation on z 2 geodesics are nearest neighbor paths in z 2 each of whose segments is time minimizing we prove part of the conjecture that doubly infinite geodesics do not exist our main tool is a result of independent interest about the coalescing of semi infinite geodesics