# Principal Mappings between Posets

Authors: Yuan Ting Nai; Dongsheng Zhao
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

We introduce and study principal mappings between posets which generalize the notion of principal elements in a multiplicative lattice, in particular, the principal ideals of a commutative ring. We also consider some weaker forms of principal mappings such as meet principal, join principal, weak meet principal,  and weak join principal mappings which also generalize the corresponding notions on elements in a multiplicative lattice, considered by Dilworth, Anderson and Johnson. The principal mappings between the lattices of powersets and chains are characterized. Finally, for any PIDR, it is proved that a mapping F:Idl(R)→Idl(R) is a contractive principal mapping if and only if there is a fixed ideal I∈Idl(R)  such that F(J)=IJ for all J∈Idl(R). This exploration also leads to some new problems on lattices and commutative rings.


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## Body

## 1. Introduction

A multiplicative lattice [1–3] is a complete lattice L together with a binary operation, called multiplication, that is associative, commutative, and distributive over arbitrary joins and has the greatest element 1L as the multiplication identity.The complete latticeIdl(R) of all ideals of a commutative ring R is a typical example of multiplicative lattices.IfI is a principal ideal of R, then I satisfies the following equations:
(1)(J∩[K:I])I=(JI)∩K,[(K+(JI)):I]=[K:I]+J,
for any J,K∈Idl(R) where [K:I] is the ideal quotient [4].In terms of the order and the multiplication on the latticeIdl(R), the above two equations can be rephrased as
(2)(J∧[K:I])I=(JI)∧K,[(K∨(JI)):I]=[K:I]∨J.In his efforts to obtain an abstract ideal theory of commutative rings, Dilworth introduced principal elements, the analogues of principal ideals, in a multiplicative lattice. The definition of principal elements makes use of the corresponding properties of principal ideals given in (2). Based on this notion of principal elements, Dilworth successfully established Noether’s Decomposition Theorems and Krull’s Principal Ideal Theorem for multiplicative lattices.Thereafter, the principal elements have been studied extensively by many people including Anderson, Johnson, and others [5–10]. As pointed out by Anderson and Johnson [2], “principal elements are the cornerstone on which the theory of multiplicative lattices and abstract ideal theory now largely rest.” Dilworth’s original definition of principal elements is only valid for a multiplicative lattice as it makes use of the multiplication, meet and join operations in a lattice, and so it does not apply on a general lattice. It is thus natural to wonder whether it is possible to extend this notion to arbitrary lattices or even posets.Let us relook at the principal ideals of a commutative ringR from the perspective of mappings. Each ideal  I  of  R defines two mappings  FI,GI:Idl(R)→Idl(R)  by
(3)FI(J)=IJ,GI(J)=[J:I],
for each  J∈Idl(R).By (2), if I is a principal ideal of R, then FI and GI satisfy the following equations for any J,K∈Idl(R):
(4)FI(J∧GI(K))=FI(J)∧K,GI(K∨FI(J))=GI(K)∨J.Thus, every principal idealI of  R corresponds to a special mapping FI from the lattice Idl(R) to itself.Motivated by these observations, we defined principal mappings between lattices and proved some basic properties of such mappings in [11]. In the current paper, we further generalize the notion of principal mappings to arbitrary posets and systematically explore their properties and investigate various examples. The introduction of principal mappings also provides a new perspective in the study of ideals of commutative rings. For example, for a given ring R, one can consider which principal mapping from Idl(R) to itself is of the form FI for some ideal I of R. The notion of principal mapping also provides a platform to compare and link the principal elements in a multiplicative lattice with the corresponding objects in other ordered structures such as the open mappings between topological spaces and the injective or surjective mappings between sets.The layout of the paper is as follows. In Section2, we define the principal mappings, meet principal and join principal mappings, between posets and prove some of their basic properties. In Section 3, we study weak principal mappings which generalize the corresponding notions of elements and their properties. In Sections 4 and 5, the principal mappings between some special types of lattices, such as the lattices of power sets and chains, are characterized. We also characterize the principal mappings f satisfying f(x)≤x (called contractive principal mappings) for some types of lattices such as the lattice of ideals of a principal ideal domain. Some open problems on principal mappings are posed.In this paper, we assume the readers have basic knowledge on lattices and multiplicative lattices. For posets, lattices and more on adjunctions, see [12–14]. For multiplicative lattices, we refer to [1–3].


## 2. Principal Mappings between Posets

Throughout this paper, for a poset(P,≤) and X⊆P, we denote the set {p∈P:p≤x for some x∈X} by ↓X. In the case X={x}, we simply write ↓x for ↓X. The sets ↑X and ↑x are defined dually.Definition 1.
A mappingf:P→Q between two posets is called aprincipal mapping if there is a mapping g:Q→P such that the following equations hold for all x∈P and y∈Q:
(5)f(↓x∩↓g(y))=↓f(x)∩↓y,(6)g(↑y∩↑f(x))=↑g(y)∩↑x.
The mapping g is then called theresidual of f.
Iff and g satisfy (5), then f is called ameet principal mapping.
Iff and g satisfy (6), then f is called ajoin principal mapping.
Thus,f is a principal mapping if and only if it is both meet principal and join principal mappings.The following are some immediate examples.Examples 1.
(1) Every isomorphismf:P→Q between posets is a principal mapping. In this case, the residual of f is the inverse mapping of f.
(2) For any setX, the power set 𝒫(X) of X is a poset. Given any A∈𝒫(X), the mapping FA:𝒫(X)→𝒫(X) is a principal mapping, where FA(B)=A∩B for any B∈𝒫(X) (see Corollary 30).
(3) Let(Idl(R),⊆) be the lattice of ideals of a commutative ring R with inclusion as the partial order. For any principal ideal I0∈Idl(R), the mapping FI0:Idl(R)→Idl(R) is a principal mapping where FI0(J)=I0J (the multiplication of I0 with J) for any J∈Idl(R). The residual of FI0 sends J to the ideal quotient [J:I0]. See Theorem 7 for the general result on lattices.
(4) LetX be a topological space and 𝒪(X) be the complete lattice of open sets of X. Given any U∈𝒪(X), define the mapping ϕU:𝒪(X)→𝒪(X) by ϕU(V)=U∩V, V∈𝒪(X). Then, ϕU is a meet principal mapping that is not a join principal mapping unless U is closed.
(5) Letf:X→Y be a continuous open mapping between topological spaces X and Y (i.e., f(U) is an open set of Y for any open set U of X). Let f*:𝒪(X)→𝒪(Y) be defined by f*(U)=f(U), U∈𝒪(X). Then, f* is a meet principal mapping. In general, f* is not a join principal mapping.More examples of principal mappings, meet principal mappings, and join principal mappings will be considered in later sections.Recall that an adjunction is a pair(f,g) of monotone mappings f:P→Q and g:Q→P between posets such that, for all  x∈P and y∈Q, f(x)≤y if and only if x≤g(y) (see Definition O-3.1 of [14]). In this case, g is called the upper adjoint of f and f is called the lower adjoint of g.Equivalently,(f,g) is an adjunction if and only if f and g are monotone and g(f(x))≥x and f(g(y))≤y hold for all x∈P, y∈Q (Theorem O-3.6 of [14]).Lemma 2.
Letf:P→Q be a principal mapping between posets with g:Q→P as the residual. Then, (f,g) is an adjunction.Proof.
Puttingx=g(y) in (5) and y=f(x) in (6) in Definition 1, we obtain
(7)f(↓g(y))=↓f(g(y))∩↓y,g(↑f(x))=↑g(f(x))∩↑x.
These imply that f(g(y))≤y and g(f(x))≥x for all x∈P and y∈Q.
For anyx1,x2∈P with x1≤x2, ↓x1⊆↓x2⊆↓g(f(x2)). As f is a principal mapping, so
(8)f(↓x1)=f(↓x1∩↓g(f(x2)))=↓f(x1)∩↓f(x2),
implying f(x1)≤f(x2), and thus f is monotone. Similarly, we can show that g is monotone. Thus, (f,g) is an adjunction.As the upper adjoint of a mapping is unique, the residual of a principal mapping is also unique.The following theorem can be proved using Definition1 and Lemma 2.Theorem 3.
Letf:P→Qbe a mapping between posets. Then, f is a principal mapping if and only if
(i)f has an upper adjoint g:Q→P;
(ii) for allx∈P and y∈Q,
(9)↓f(x)=f(↓x),↑g(y)=g(↑y).Remark 4.
Iff:P→Q has an upper adjoint g:Q→P, then both f and g are monotone. Hence, for any x∈P, y∈Q and f(↓x)⊆↓f(x), g(↑y)⊆↑g(y). Therefore, to prove that f is a principal mapping, it suffices to show that ↓f(x)⊆f(↓x) and ↑g(y)⊆g(↑y).
In particular, to prove thatf is a meet (join) principal mapping, it suffices to show that ↓f(x)⊆f(↓x)(↑g(y)⊆g(↑y)).The example below gives a lower adjoint mapping that is not principal.Example 5.
Let[0,1] be the unit interval of real numbers with the usual order of numbers. Define f:[0,1]→[0,1] by
(10)f(x)={0,x=0,1,x≠0.
Then,f has an upper adjoint g:[0,1]→[0,1] given by g(1)=1, g(x)=0(x≠1). But ↓f(1)≠f(↓1); hence, f is not a principal mapping.In the case whereP and Q are semilattices or lattices, we have a neater characterization of the types of mappings defined in Definition 1.Theorem 6.
Letf:P→Q be a mapping between posets with an upper adjoint.
(i) IfP is a meet semilattice, then f is a meet principal mapping if and only if, for all x∈P and y∈Q,
(11)f(x∧g(y))=f(x)∧y.
(ii) IfQ is a join semilattice, then f is a join principal mapping if and only if, for all x∈P and y∈Q,
(12)g(y∨f(x))=g(y)∨x.Theorem 7.
Letf:L→M be a mapping between two lattices. Then, f is a principal mapping if and only if there is a mapping g:M→L such that, for all x∈L, y∈M,
(13)f(x∧g(y))=f(x)∧y,g(y∨f(x))=g(y)∨x.The following proposition easily follows from the definition of principal mappings.Proposition 8.
If
f
1
:
P
1
→
P
2
and
f
2
:
P
2
→
P
3
are principal mappings between posets, then
f
2
∘
f
1
:
P
1
→
P
3
is a principal mapping.For any two elementsx and y in a multiplicative lattice L, the residual [y:x] is defined as
(14)[y:x]=⋁{b∈L:bx≤y}.Given an elementa in a multiplicative lattice L, let ϕa:L→L be the mapping such that ϕa(x)=ax and ϕa:L→L be the mapping given by
(15)ϕa(y)=⋁{x∈L:ax≤y}=[y:a].
Then, (ϕa,ϕa) forms an adjunction.By Dilworth [1], an element a of a multiplicative lattice L is called a principal element if and only if, for any x,y∈L,
(16)(x∧[y:a])a=xa∧y,[(y∨xa):a]=[y:a]∨x.The elementa is called a meet principal element if it satisfies (x∧[y:a])a=xa∧y and a join principal element if it satisfies [(y∨xa):a]=[y:a]∨x for any x,y∈L.The above two equations can be rewritten in terms ofϕa and ϕa as follows:
(17)ϕa(x∧ϕa(y))=ϕa(x)∧y,ϕa(y∨ϕa(x))=ϕa(y)∨x.Therefore, by Theorem7, we have the following.Proposition 9.
(i) An element
a
of a multiplicative lattice
L
is a meet (join) principal element if and only if the mapping
ϕ
a
:
L
→
L
is meet (join) principal; (ii)
a
is a principal element if and only if the mapping
ϕ
a
:
L
→
L
is principal.Applying Theorem3 to the mappings ϕa and ϕa defined above, we deduce the following result where (i) appeared in [2].Corollary 10.
LetL be a multiplicative lattice and a∈L.
(i)a is meet principal if and only if, for any x,y∈L with y≤xa, there is p∈L and p≤x such that y=pa.
(ii)a is join principal if and only if, for any x,y∈L with x≥[y:a], there is q∈L and q≥y such that x=[q:a].Note that for any two elementsa and b in a multiplicative lattice L, ϕa∘ϕb=ϕab. Thus, by Propositions 8 and 9, we deduce the following result which first appeared in [1].Corollary 11.
The product of two principal elements of a multiplicative lattice is a principal element.Remark 12.
LetPRMP(L,L) be the set of all principal mappings from the multiplicative lattice L to itself, and let PRE(L) be the set of all principal elements of L. Then, (PRMP(L,L),∘) (∘ denotes the composition operation) is a semigroup with idL as the identity, and (PRE(L),·) is also a semigroup with 1L as the identity, and · as the multiplication on L. Now the mapping a↦ϕa defines an embedding of (PRE(L),·) into (PRMP(L,L),∘).In a multiplicative latticeL,ab≤b holds for any elements a,b. Hence, the mapping ϕa:L→L satisfies the following condition:
(18)ϕa(x)≤x∀x∈L.
This is equivalent to
(19)x≤ϕa(x)=[x:a]∀x∈L.For any posetP, a mapping f:P→P will be calledcontractive if f(x)≤x for all x∈P. If f has an upper adjoint g:P→P, then f is contractive if and only if x≤g(x) for all x∈P.A mappingf:P→P from a poset to itself is called acontractive principal mapping if f is both principal and contractive.One of the problems on multiplicative lattices (in particular,Idl(R) for a commutative ring R) we shall address is as follows. Given a multiplicative lattice L, under what condition every contractive principal mapping f:L→L is of the form ϕa for some principal element a∈L?


## 3. Weak Principal Mappings between Posets

In this section, we consider weak principal mappings and their some links to modular lattices.LetL be a multiplicative lattice with bottom element 0L and top element 1L. By [2], an element a of L is called(i) a weak meet principal element if, for allx∈L,
(20)[x:a]a=a∧x;(ii) a weak join principal element if, for allx∈L,
(21)[xa:a]=[0L:a]∨x.We now define the corresponding notions for mappings between posets, whose definitions appear more natural than the corresponding ones for elements.Definition 13.
Letf:P→Q be a mapping between posets with g:Q→P as the upper adjoint of f.
(i)f is called aweak meet principal mapping if
(22)↓f(P)=f(P).
(ii)f is called aweak join principal mapping if
(23)↑g(Q)=g(Q).
Iff is both a weak meet principal mapping and a weak join principal mapping, then f is called aweak principal mapping.Remark 14.
(1) To prove thatf is a weak meet (weak join) principal mapping, it suffices to show that ↓f(P)⊆f(P)(↑g(Q)⊆g(Q),resp.).
(2) Iff:P→Q has an upper adjoint g:Q→P and P has a top element 1P (Q has a bottom element 0Q), then f is weak meet (weak join) principal if and only if ↓f(1P) = f(↓1P) = f(P)(↑g(0Q)=g(↑0Q)=g(Q)).Examples 2.
(1) LetL be a frame with 0L and 1L as the bottom and top elements, respectively. For any a∈L, the mapping ψa:L→L, defined by ψa(x)=a∧x  for all  x∈L, is always a weak meet principal mapping. ψa is a weak join principal mapping if and only if there exists a⊥∈L such that a∧a⊥=0L and a∨a⊥=1L  (i.e., a has a complement).
(2) Leth:X→Y be a mapping from set X to set Y. Then, the mapping Fh:𝒫(Y)→𝒫(X) between lattices of power sets of Y and X, respectively, defined by Fh(V)=h-1(V) for any V∈𝒫(Y), is a weak join principal mapping. Fh is a weak meet principal mapping if and only if h is injective.Proposition 15.
Let
f
:
P
→
Q
be a mapping between posets with
g
:
Q
→
P
as the upper adjoint.
(i) If
P
has a top element
1
P
, then
f
is a weak meet principal mapping if and only if, for all
y
∈
Q
,
(24)
f
(
g
(
y
)
)
=
f
(
1
P
)
∧
y
.
(ii) If
Q
has a bottom element
0
Q
, then
f
is a weak join principal mapping if and only if, for all
x
∈
P
,
(25)
g
(
f
(
x
)
)
=
g
(
0
Q
)
∨
x
.Proof.
(i) Supposef is a weak meet principal mapping. Clearly, for any y∈Q,f(g(y))  is a lower bound of {f(1P),y}. Now, if u is a lower bound of {f(1P),y}, then u∈↓f(P)=f(P), implying that u=f(z) for some z∈P which implies that z≤g(y). Hence, u=f(z)≤f(g(y)) and so f(g(y))=f(1P)∧y.
Conversely, supposef(g(y))=f(1P)∧yholds for all y∈Q. For any v∈↓f(P), there exists a∈P such that v≤f(a)≤f(1P). So, v=f(1P)∧v=f(g(v))∈f(P). Thus, ↓f(P)⊆f(P), so f is a weak meet principal.
(ii) follows a dual proof to that of (i).Using the relation between an elementa in a multiplicative lattice L and the mapping ϕa:L→L defined by ϕa(x)=ax for all x∈L, we obtain the following results.Corollary 16.
An elementa in a multiplicative lattice L is weak meet (weak join) principal if and only if the mapping ϕa:L→L is weak meet (weak join) principal.Corollary 17.
An elementa in a multiplicative lattice L is
(i) weak meet principal if and only if, for anyy∈L with y≤a, there is p∈L such that y=pa;
(ii) weak join principal if and only if, for anyx∈L with x≥[0:a], there is q∈L such that x=[q:a].Corollary17 (i) is Lemma  1 (a) in [2].Remark 18.
Iff:P→Q is a meet principal mapping between posets, then f(↓x)=↓f(x) holds for all x∈P, and so ↓f(P)=f(P). Thus, a meet principal mapping is weak meet principal. Similarly, every join principal mapping between posets is weak join principal.A weak meet (weak join) principal mapping need not be meet (join) principal. A counterexample can be easily constructed by considering the mappings from the nonmodular lattice, the pentagonN5, to itself.Recall that a latticeL is said to bemodular if, for any x,a,b∈L with b≤x,  x∧(a∨b)=(x∧a)∨b  (see, e.g., [12, 13]).Proposition 19.
Let
f
:
L
→
M
be a mapping between bounded lattices
L
and
M
with an upper adjoint such that
f
is a weak principal mapping.
(i) If
L
is modular, then
f
is a meet principal mapping.
(ii) If
M
is modular, then
f
is a join principal mapping.Proof.
Letg:M→L be the upper adjoint of f.
(i) For anya∈M with a≤f(b) for some b∈L, we have u=g(a)≤g(f(b))=b∨g(0M) since f is weak join principal. Also, a≤f(b)≤f(1L) which implies that a=a∧f(1L)=f(g(a))=f(u) as f is weak meet principal. Since u≤b∨g(0M), L is modular and u=g(a)≥g(0M),
(26)f(u)=f(u∧(b∨g(0M)))=f((u∧b)∨g(0M)).
As a lower adjoint, f preserves joins. Furthermore, f(g(0M))≤0M which implies that  f(g(0M))=0M. Now,
(27)a=f(u)=f(u∧b)∨f(g(0M))=f(u∧b)∨0M=f(u∧b)∈f(↓b).
Hence, f is meet principal.
(ii) follows a proof dual to that of (i).The following theorem now follows from Proposition19.Theorem 20.
Letf:L→M be a mapping between bounded modular lattices with an upper adjoint. Then, f is a principal mapping if and only if f is a weak principal mapping.Corollary 21.
IfL is a bounded modular lattice, then every weak principal mapping f:L→L is principal.Using Theorem20, Proposition 9, and Corollary 16, we obtain the following result in [2, 7].Corollary 22.
An element in a modular multiplicative latticeL is principal if and only if it is weak principal.A natural question arising here is whether the converse of Corollary21 is true. If L is a bounded lattice and every weak principal mapping f:L→L is principal, must  L be modular?Example 23.
Consider the latticeL={0,a,b,c,e,f,g,1} where 0<a<b<d,  0<c<d<e<1, and d<f<1. One can check that every weak principal mapping f:L→L is principal. However,  L  is not modular.
Another counterexample, suggested by Dr. Peter Jipsen, is the 6-element nonmodular latticeH={0,a,b,c,d,1} where 0<a<d<1,  0<b<d,  and  0<c<1.The following problem is still open.Problem 24.
LetLbe a bounded lattice such that for any bounded lattice M, every weak principal mapping f:L→M is meet principal. Must L be modular?The composition of two weak meet (weak join) principal mappings need not be weak meet (weak join) principal. A counterexample can be easily constructed by considering the composition of two weak meet principal mappings from the nonmodular lattice, the pentagonN5, to itself.Proposition 25.
Let
f
1
:
P
1
→
P
2
and
f
2
:
P
2
→
P
3
be mappings between posets.
(1) If
f
1
is weak meet principal and
f
2
is meet principal, then
f
2
∘
f
1
:
P
1
→
P
3
is weak meet principal.
(2) If
f
1
is join principal and
f
2
is weak join principal, then
f
2
∘
f
1
:
P
1
→
P
3
is weak join principal.Using Propositions9 and 25 and Corollary 16, we obtain the following result which is Proposition 1  (a) in [2].Proposition 26.
If
d
is a weak meet principal element and
e
is a meet principal element in a multiplicative lattice
L
, then
d
e
is a weak meet principal element in
L
.For bounded semilattices, we have another characterization of weak join principal and weak meet principal mappings.Proposition 27.
Let
f
:
L
→
M
be a mapping between two posets with
g
:
M
→
L
as the upper adjoint of
f
.
(1) If
L
and
M
are join semilattices with
0
M
as the bottom element of
M
, then
f
is a weak join principal mapping if and only if, for any
b
,
c
∈
L, f(b)≤f(c)implies thatb≤c∨g(0M).
(2) If
L
and
M
are meet semilattices with
1
L
as the top element of
L
, then
f
is a weak meet principal mapping if and only if, for any
a
,
d
∈
M
,
g
(
a
)
≥
g
(
d
)
implies that
a
≥
d
∧
f
(
1
L
)
.Proof.
As before, we just gave the proof of (1).
Supposef is weak join principal and f(b)≤f(c) where b,c∈L. Then, b≤g(f(b))≤g(f(c))=c∨g(0M).
Conversely, supposef:L→M satisfies the given condition. Clearly, b∨g(0M)≤g(f(b)) for any b∈L. For any b∈L, f(g(f(b)))≤f(b), so, by the assumption on f, we have g(f(b))≤b∨g(0M).By [14], a lower adjoint is injective if and only if its upper adjoint is surjective. Now, if f:L→M is an injective weak join principal mapping between two join semilattices with bottom elements 0L and 0M,  respectively, then we must have g(0M)=0L, where g:M→L is the upper adjoint of  f. Then, f(b)≤f(c) implies that b≤c∨g(0M)=c∨0L=c, so f is an order embedding.Corollary 28.
Iff:L→M is an injective weak join principal mapping between join semilattices with bottom elements, then f is an order embedding.


## 4. Principal Mappings between Lattices of Powersets and Chains

In this section, we investigate the principal mappings and their weaker versions between the lattices of powersets and chains.For any setX, the power set 𝒫(X) of X is a complete lattice with respect to the inclusion order.Theorem 29.
LetX and Y be two nonempty sets. For any mapping F:𝒫(X)→𝒫(Y), the following statements are equivalent.
(1)F is a principal mapping.
(2) There is a subsetX1 of X and an injective mapping t:X1→Y such that F(A)=t(A∩X1) for all A∈𝒫(X).Proof.
(1) implies (2). LetF be principal and let G be its upper adjoint. As every lower adjoint preserves arbitrary joins, it sends bottom element to the bottom element. Thus, F(∅)=∅.
(i) For anyx∈X, if F({x}) is nonempty, then it is a singleton set. As a matter of fact, assume u∈F({x}). Then, {u}∈↓F({x}), where ↓F({x})={A⊆X:A⊆F({x})}. Since F is meet principal, there is a subset B⊆{x} such that F(B)={u}. Clearly, B≠∅, so B={x}, which implies that F({x})={u} is a singleton set.
(ii) LetX1={x∈X:F({x})≠∅}. Then, F(X-X1)=⋃‍{F({z}):z∈X-X1}=∅ because F preserves arbitrary joins. It then follows that G(∅)=X-X1. Now, by Proposition 15 (ii), for any u∈X1, G(F({u}))=G(∅)∪{u}=(X-X1)∪{u}. Thus, u≠v and u, v∈X1 imply that F({u})≠F({v}).
(iii) Definet:X1→Y by t(x)=y, where {y}=F({x}). By (ii), t is an injective mapping. For any A∈𝒫(X), note that F preserves joins (i.e., unions in this case), so F(A)=F(A-X1)∪F(A∩X1)=∅∪t(A∩X1).
(2) implies (1). Assume the condition in (2) is satisfied. Then,F has an upper adjoint G given by G(B)=t-1(B)∪(X-X1) for each B∈𝒫(Y). For any A⊆XandB⊆Y, if y∈F(A)∩B, then y=t(x) for some x∈A∩X1. Also, y=t(x)∈B which implies that x∈t-1(B)⊆G(B). It follows that y=t(x)∈F(G(B)∩A). Hence, F(G(B)∩A)=F(A)∩B.
Ifx∈G(B∪F(A)), then F({x})⊆B∪F(A). But F({x})=t({x}∩X1) is at most a nonempty singleton set. If F({x})⊆B, then x∈G(B). If F({x})⊈B, then F({x})≠∅ and F({x})=t({x}∩X1)⊆F(A)=t(A∩X1), implying that x∈A since t is injective. It follows that G(B∪F(A))=G(B)∪A. Therefore, F is principal.For any subsetX1 of a set X, let t:X1→X be the embedding mapping; that is, t(x)=x, for any x∈X1. Then, for any A∈𝒫(X), t(A∩X1)=A∩X1. By Theorem 29 (2), we have the following.Corollary 30.
For any fixed subsetX1 of a set X, the mapping FX1:𝒫(X)→𝒫(X) is a principal mapping, where FX1(B)=X1∩B for any B⊆X.Corollary 31.
Letf:X→Y be a mapping between nonempty sets X and Y. Then, the mapping F:𝒫(X)→𝒫(Y), defined by F(A)=f(A)={f(x):x∈A} for any A∈𝒫(X), is a principal mapping if and only if f is injective.For contractive principal mappings on lattices of powersets, we have the following.Proposition 32.
A mappingF:𝒫(X)→𝒫(X) is a contractive principal mapping if and only if there exists a subset X1 of X such that F(A)=A∩X1 for all A∈𝒫(X).Proof.
Assume thatF is a principal mapping that is also contractive. By Theorem 29, there is a subset X1 of X and an injective mapping t:X1→X such that F(A)=t(A∩X1) for all A∈𝒫(X). Since F is contractive, for any x∈X1,  F({x})=t({x}∩X1)⊆{x} which implies that t(x)=x for all x∈X1. Hence, F(A)=A∩X1  for all A∈𝒫(X).Problem 33.
Determine the complete latticesL such that for any contractive principal mapping F:L→L, there is an element a∈L such that F(x)=a∧x for all x∈L.Recall that a chain is a poset in which every two elements are comparable.Theorem 34.
LetC be a chain and let f:C→C be a mapping. Then, the following statements are equivalent.
(1)f is a principal mapping.
(2)f is a meet principal mapping and, for any x,y∈C,  x<y and f(x)=f(y) imply that f(z)=f(y) for all z≤y.
IfC does not have a bottom element, then (2) is equivalent to the following.
(3)f is a meet principal mapping that is strictly monotone.Proof.
(1) implies (2). Letf be principal and let g be its residual. Suppose x,y∈C such that x<y and f(x)=f(y). Clearly, for any z with  x<z<y, f(z)=f(y).
If there exists az0∈C such that z0≤x<y, and f(x)≠f(z0), then, as f is monotone, we have f(z0)<f(x), which implies that g(f(z0))≱x; hence, g(f(z0))<x. Thus, x∈↑g(f(z0))=g(↑f(z0)). So there exists k≥f(z0) such that x=g(k). Then, f(y)=f(x)=f(g(k))≤k, so y≤g(k)=x, which contradicts x<y. Hence, for all z≤x<y, we have f(z)=f(x)=f(y).
(2) implies (1). We only need to show thatf is join principal. Let g be the upper adjoint of f. For any a∈C, let y>g(a). Then, f(y)≥f(g(a)). If f(y)=f(g(a)), then y≤g(f(y))=g(f(g(a)))=g(a) (note that, for any adjunction (f,g),g∘f∘g=g and f∘g∘f=f), which contradicts y>g(a). Hence, f(y)>f(g(a)). Also, if f(y)≤a, then y≤g(a) which contradicts y>g(a) again. It follows that a∈↓f(y)=f(↓y) since f is a meet principal mapping. There exists u∈↓y such that a=f(u). Hence, f(g(a)) = f(g(f(u))) = f(u)=a.
We now show thatg(f(y))=y. For this, we only need to show y≥g(f(y)) as g(f(y))≥y always holds. Suppose y≱g(f(y)); then y<g(f(y)). As f(y)=f(g(f(y))) and g(a)<y<g(f(y)), so by the assumption in (2), a=f(g(a))=f(g(f(y)))=f(y), a contradiction. It follows that y=g(f(y)) and f(y)∈↑a. Thus, ↑g(a)⊆g(↑a) and so f is join principal. Hence, f is principal.
Now assumeC does not have a bottom element and that (3) implying (2) is trivial.
(2) implies (3). Supposef is not strictly monotone. Then, there exist a,b∈C, a≠b, such that f(a)=f(b). Without loss of generality, assume that a<b. By (2), for all x≤b, f(x)=f(b). Since C does not have a bottom element, there exists y0∈C such that y0<f(b). So, y0∈↓f(b)=f(↓b) since f is a meet principal mapping. Thus, there exists x0≤b such that y0=f(x0). But then f(x0)=f(b), and we have y0=f(b), which contradicts y0<f(b). Hence, f must be strictly monotone.LetC={c1,…,cn} be a finite chain where c1<c2<⋯<cn. For each k, 1≤k≤n, let fk:C→C be the mapping given by fk(cn)=ck, fk(cn-1)=ck-1,…,fk(cn-k+1)=fk(cn-k)=⋯=fk(c1)=c1. Then, using Theorem 34, one can verify that fk is a principal mapping. Conversely, if f:C→C is a principal mapping, then  f=fk, where  f(cn)=ck.Corollary 35.
For any chainC with n elements, there are exactly n principal mappings f:C→C.One of the main tasks we are interested in is to determine all the principal mappingsf:Idl(R)→Idl(R)  where  Idl(R) is the lattice of ideals of a commutative ring R.Corollary 36.
For any primep and n∈ℤ+, there are exactly n+1 principal mappings f:Idl(ℤpn)→Idl(ℤpn) (ℤpn denotes the ring of integers modulo pn). Thus, for every principal mapping f:Idl(ℤpn)→Idl(ℤpn), there exists  I∈Idl(ℤpn) such that f=FI, where FI(J)=IJ for any J∈Idl(ℤpn).Proof.
(
ℤ
p
n
,
+
,
·
) is a principal ideal domain and there are exactly n+1 ideals forming a chain. So the first part follows from Corollary 35. For the second part, just note that each FI(I∈Idl(ℤpn)) is principal and that I≠K for I and K∈Idl(ℤpn) implies that FI≠FK.An elementa of a complete lattice L is called apseudoprincipal element if there is a contractive principal mapping f:L→L such that  f(1L)=a.Ifa is a principal element of a multiplicative lattice L, then a=ϕa(1L), so a is pseudoprincipal. The proof of Corollary 35 also shows that every element in a finite chain is pseudoprincipal.


## 5. Principal Mappings between Lattices of Ideals of a Principal Ideal Domain

We now investigate the principal mappings between the complete lattices of ideals of some special types of commutative rings, in particular, principal ideal domains. The main purpose is to identify those principal mappings that are defined by a principal ideal.For any idealI of a commutative ring R, the mapping FI:Idl(R)→Idl(R) given by FI(J)=IJ for all J∈Idl(R) will be called themapping defined byI. Note that FI is always a lower adjoint of the mapping GI:Idl(R)→Idl(R) defined by GI(K)=[K:I] for any K∈Idl(R). By Proposition 9 (ii), FI is a principal mapping if and only if I is a principal element of the multiplicative lattice Idl(R).Definition 37.
A multiplicative latticeL has thedivisibility order if, for all  a,b∈L and  a≤b if and only if b divides a (i.e., a=bc for some c∈L).
Recall that acancellation ideal ring [15] is a commutative ring R in which every nonzero ideal I of R is a cancellation ideal [16]; that is, whenever IJ=IK for any J,K∈Idl(R), and then J=K. It is easy to see that the above condition that is equivalent to IJ⊆IK implies that J⊆K.
Note that every principal ideal domainR is a cancellation ideal ring and the corresponding multiplicative lattice Idl(R) has the divisibility order.Theorem 38.
LetR be a cancellation ideal ring such that Idl(R) has the divisibility order. Then, ϕ:Idl(R)→Idl(R) is an injective principal mapping if and only if there is an order isomorphism ψ:Idl(R)→Idl(R) and a nonzero principal element I of Idl(R) such that ϕ=FI∘ψ.Proof.
Only the necessity needs verification because every isomorphism is a principal mapping and the composition of principal mappings is principal.
Letϕ:Idl(R)→Idl(R) be an injective principal mapping and let ϕ* be its upper adjoint. Since ϕ is injective, ϕ(R)≠{0R}. For any J∈Idl(R), since ϕ(J)⊆ϕ(R) and Idl(R) has the divisibility order, there exists H∈Idl(R) such that ϕ(J)=Hϕ(R). As ϕ(R)≠{0R}, R is a cancellation ideal ring, such an H is unique for each J, which will be denoted by ψ(J). Now, ϕ(J)=ψ(J)ϕ(R)=(Fϕ(R)∘ψ)(J).
The mappingψ is injective because ϕ is injective. For any I∈Idl(R), since Iϕ(R)⊆ϕ(R) and ϕ is a principal mapping, by Theorem 3, there exists J∈Idl(R) such that ϕ(J)=Iϕ(R). But ϕ(J)=ψ(J)ϕ(R). So ψ(J)ϕ(R)=Iϕ(R), which implies that I=ψ(J) as R is a cancellation ideal ring and so ψ is surjective. Suppose ψ(J)⊆ψ(K). Then, ϕ(J)=ϕ(R)ψ(J)⊆ϕ(R)ψ(K)=ϕ(K). It follows that
(28)J⊆ϕ*(ϕ(J))⊆ϕ*(ϕ(K))=K∨ϕ*({0R})=K;
the first equality follows from that ϕ is join principal and Proposition 15 (ii) and the second equality follows from the assumption that ϕ is injective (so its upper adjoint ϕ* is surjective implying ϕ*{0R}={0R}). Thus, J⊆K.
We now show thatψ is monotone. For any J,K∈Idl(R) such that J⊆K, since ϕ is monotone, ϕ(J)⊆ϕ(K). So  ψ(J)ϕ(R)⊆ψ(K)ϕ(R) and since ϕ(R)≠{0R}, it follows that ψ(J)⊆ψ(K) as R  is a cancellation ideal ring. Hence, ψ is indeed an order isomorphism.
Finally,Fϕ(R)=ϕ∘ψ-1, as a composition of principal mappings is principal; thus ϕ(R) is a principal element of Idl(R).In the following, for any elementx of a commutative ring, 〈x〉 denotes the principal ideal generated by x.The following fact will be used several times in later arguments.Lemma 39.
LetR be a principal ideal domain. For any nonzero a∈R, any prime element p in R and t∈ℤ+〈pt〉⊆〈a〉 if and only if a=ups for some unit u and some 0≤s≤t.Recall that an elementa of a lattice L is called a (meet) irreducible element if, for any u,v∈L, a=u∧v implies that a=u  or  a=v.Lemma 40.
LetR be a principal ideal domain. Then, 〈pt〉, where p is a prime element in R and t∈ℤ+, are exactly the irreducible elements of the lattice Idl(R).Proof.
Letp be a prime element in R and t∈ℤ+. If 〈pt〉=〈a〉∩〈b〉, then, by Lemma 39, a=ups and b=vpk for some s≤t and k≤t and units u,v∈R. Thus, 〈pt〉=〈ps〉∩〈pk〉. But clearly 〈ps〉∩〈pk〉=〈pl〉, where l=max⁡{k,s}. Therefore, 〈pt〉=〈ps〉=〈a〉 or 〈pt〉=〈pk〉=〈b〉, showing that 〈pt〉 is an irreducible element of the lattice Idl(R).
Conversely, let〈a〉 be an irreducible element of the lattice Idl(R) and let a=up1t1p2t2…pktk be a prime factorization of a (note that, in any UFD, in particular, a PID, prime elements coincide with (ring) irreducible elements), where u is a unit and pi are distinct prime elements of R. Then, 〈a〉=〈p1t1〉∩〈p2t2〉∩⋯∩〈pktk〉. Thus, 〈a〉=〈piti〉 for some 1≤i≤k.Lemma 41.
LetR be a principal ideal domain. If ψ:Idl(R)→Idl(R) is an order isomorphism, then, for any prime element  p of  R, there is a prime element q of R such that for any k∈ℤ+, ψ(〈pk〉)=〈qk〉.Proof.
Since〈pk〉 is an irreducible element of the lattice Idl(R), ψ(〈pk〉) is also an irreducible element, and so, by Lemma 40, ψ(〈pk〉)=〈qm〉 for some prime element q and some  m∈ℤ+.
By Lemma39, in Idl(R), there are k-1 distinct elements lying between 〈pk〉  and 〈1〉; that is, 〈pk〉⊆〈pk-1〉⊆⋯⊆〈p2〉⊆〈p〉⊆〈1〉. Since ψ is an isomorphism, there are also k-1 distinct elements lying between ψ(〈pk〉)=〈qm〉 and  ψ(〈1〉)=〈1〉, which deduces that m=k. It is clear that the prime element q is uniquely determined by  p.We are now able to prove the main result of this section.Theorem 42.
LetR be a principal ideal domain. Then, ϕ:Idl(R)→Idl(R) is an injective contractive principal mapping if and only if ϕ=FI for some nonzero I∈Idl(R).Proof.
By Theorem38, there exist an order isomorphism  ψ:Idl(R)→Idl(R)  and a nonzero ideal I of R such that  ϕ=FI∘ψ  where FI is the mapping defined by I.
(1) We first show thatψ(〈p〉)=〈p〉 for any prime element p∈R. By Lemma 41, ψ(〈p〉)=〈q〉 for some prime element q.
Sinceϕ(〈p〉)⊆〈p〉, so Iψ(〈p〉)⊆〈p〉. As 〈p〉 is a prime ideal, either ψ(〈p〉)⊆〈p〉 or I⊆〈p〉.
(a) Ifψ(〈p〉)⊆〈p〉 holds, then 〈q〉⊆〈p〉, which implies that p=q and so ψ(〈p〉)=〈p〉.
(b) Now assume thatψ(〈p〉)⊈〈p〉. Then, I⊆〈p〉 and so I=〈cpk〉 for some k∈ℤ+, and c∈R with c≠0R such that p does not divide c.  Now, ψ(〈pk+1〉)=〈qk+1〉 by Lemma 41. Since ϕ is contractive, ϕ(〈pk+1〉)⊆〈pk+1〉. Then, 〈pk+1〉 divides ϕ(〈pk+1〉)=Iψ(〈pk+1〉)=〈cpk〉〈qk+1〉=〈cpkqk+1〉, which is not possible because p≠q and p does not divide c.
From both (a) and (b), we can deduce thatψ(〈p〉)=〈p〉.
(2) By Lemma41 and part  (1), for any  t∈ℤ+, ψ(〈pt〉)=〈pt〉.
(3) Now, for anyJ∈Idl(R), with J≠{0R} and J≠R, J=〈p1j1p2j2…ptjt〉 for some distinct prime elements pi and ji∈ℤ+, i∈{1,2,3,…,t}, for some t∈ℤ+. Consider
(29)ψ(J)=ψ〈p1j1p2j2…ptjt〉=ψ(〈p1j1〉∩〈p2j2〉∩⋯∩〈ptjt〉)=ψ(〈p1j1〉)∩ψ(〈p2j2〉)∩⋯∩ψ(〈ptjt〉)=〈p1j1〉∩〈p2j2〉∩⋯∩〈ptjt〉=J.
Also ψ(R)=R and ψ({0R})={0R} because every isomorphism sends the bottom (top) element to the bottom (top) element. It follows that ψ(J)=J for all  J∈Idl(R) and hence ϕ=FI.Problem 43.
Does Theorem42 hold for every commutative cancellation ideal ring R where Idl(R) has the divisibility order?


## 6. Conclusion and Future Works

In this paper, we introduce and study the principal mappings between posets which generalize the notion of principal elements in multiplicative lattices. A number of results on principal elements in multiplicative lattices have been generalized for such mappings. The principal mappings between some special posets have been characterized.Besides the three concrete problems posed above, the following are some more general problems for further studies.(1) Given a commutative ringR, determine all the principal mappings from the lattice Idl(R) to itself.(2) Determine, for which ringR, every injective contractive principal mapping from Idl(R) to itself is defined by an ideal of R.Letℤ[X] be the polynomial ring over the ring of integers. Is there an injective contractive principal mapping f:Idl(ℤ[X])→Idl(ℤ[X]) that is not defined by any ideal of ℤ[X]?(3)  Is it true that for any semiringR, if every weak principal mapping f:Idl(R)→Idl(R) is principal, then the lattice Idl(R) of all ideals of R is modular?



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