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 PID , it is proved that a mapping is a contractive principal mapping if and only if there is a fixed ideal such that for all . This exploration also leads to some new problems on lattices and commutative rings.

1. Introduction

A multiplicative lattice [1–3] is a complete lattice together with a binary operation, called multiplication, that is associative, commutative, and distributive over arbitrary joins and has the greatest element as the multiplication identity.

The complete lattice of all ideals of a commutative ring is a typical example of multiplicative lattices.

If is a principal ideal of , then satisfies the following equations: for any where is the ideal quotient [4].

In terms of the order and the multiplication on the lattice , the above two equations can be rephrased as

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 ring from the perspective of mappings. Each ideal    of   defines two mappings    by for each  .

By (2), if is a principal ideal of , then and satisfy the following equations for any :

Thus, every principal ideal of   corresponds to a special mapping from the lattice 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 , one can consider which principal mapping from to itself is of the form for some ideal of . 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 Section 2, 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 satisfying (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 and , we denote the set for some by . In the case , we simply write for . The sets and are defined dually.

Definition 1. A mapping between two posets is called a principal mapping if there is a mapping such that the following equations hold for all and : The mapping is then called the residual of .
If and satisfy (5), then is called a meet principal mapping.
If and satisfy (6), then is called a join principal mapping.
Thus, 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 isomorphism between posets is a principal mapping. In this case, the residual of is the inverse mapping of .
(2) For any set , the power set of is a poset. Given any , the mapping is a principal mapping, where for any (see Corollary 30).
(3) Let be the lattice of ideals of a commutative ring with inclusion as the partial order. For any principal ideal , the mapping is a principal mapping where (the multiplication of with ) for any . The residual of sends to the ideal quotient . See Theorem 7 for the general result on lattices.
(4) Let be a topological space and be the complete lattice of open sets of . Given any , define the mapping by , . Then, is a meet principal mapping that is not a join principal mapping unless is closed.
(5) Let be a continuous open mapping between topological spaces and (i.e., is an open set of for any open set of ). Let be defined by , . Then, is a meet principal mapping. In general, 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 of monotone mappings and between posets such that, for all   and , if and only if (see Definition O-3.1 of [14]). In this case, is called the upper adjoint of and is called the lower adjoint of .

Equivalently, is an adjunction if and only if and are monotone and and hold for all , (Theorem O-3.6 of [14]).

Lemma 2. Let  be a principal mapping between posets with as the residual. Then, is an adjunction.

Proof. Putting in (5) and in (6) in Definition 1, we obtain These imply that and for all and .
For any with , . As is a principal mapping, so implying , and thus is monotone. Similarly, we can show that is monotone. Thus, 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 Definition 1 and Lemma 2.

Theorem 3. Let be a mapping between posets. Then, is a principal mapping if and only if
(i) has an upper adjoint ;
(ii) for all and ,

Remark 4. If has an upper adjoint , then both and are monotone. Hence, for any , and , . Therefore, to prove that is a principal mapping, it suffices to show that and .
In particular, to prove that is a meet (join) principal mapping, it suffices to show that .

The example below gives a lower adjoint mapping that is not principal.

Example 5. Let be the unit interval of real numbers with the usual order of numbers. Define by
Then, has an upper adjoint given by , . But ; hence, is not a principal mapping.

In the case where and are semilattices or lattices, we have a neater characterization of the types of mappings defined in Definition 1.

Theorem 6. Let be a mapping between posets with an upper adjoint.
(i) If is a meet semilattice, then is a meet principal mapping if and only if, for all and ,
(ii) If is a join semilattice, then is a join principal mapping if and only if, for all and ,

Theorem 7. Let be a mapping between two lattices. Then, is a principal mapping if and only if there is a mapping such that, for all , ,

The following proposition easily follows from the definition of principal mappings.

Proposition 8. If and are principal mappings between posets, then is a principal mapping.

For any two elements and in a multiplicative lattice , the residual is defined as

Given an element in a multiplicative lattice , let be the mapping such that and be the mapping given by Then, forms an adjunction.

By Dilworth [1], an element of a multiplicative lattice is called a principal element if and only if, for any ,

The element is called a meet principal element if it satisfies and a join principal element if it satisfies for any .

The above two equations can be rewritten in terms of and as follows:

Therefore, by Theorem 7, we have the following.

Proposition 9. (i) An element of a multiplicative lattice is a meet (join) principal element if and only if the mapping is meet (join) principal; (ii) is a principal element if and only if the mapping is principal.

Applying Theorem 3 to the mappings and defined above, we deduce the following result where (i) appeared in [2].

Corollary 10. Let be a multiplicative lattice and .
(i) is meet principal if and only if, for any with , there is and such that .
(ii) is join principal if and only if, for any with , there is and such that .

Note that for any two elements and in a multiplicative lattice , . 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. Let be the set of all principal mappings from the multiplicative lattice to itself, and let be the set of all principal elements of . Then, ( denotes the composition operation) is a semigroup with as the identity, and is also a semigroup with as the identity, and as the multiplication on . Now the mapping defines an embedding of into .

In a multiplicative lattice holds for any elements . Hence, the mapping satisfies the following condition: This is equivalent to

For any poset , a mapping will be called contractive if for all . If has an upper adjoint , then is contractive if and only if for all .

A mapping from a poset to itself is called a contractive principal mapping if is both principal and contractive.

One of the problems on multiplicative lattices (in particular, for a commutative ring ) we shall address is as follows. Given a multiplicative lattice , under what condition every contractive principal mapping is of the form for some principal element ?

3. Weak Principal Mappings between Posets

In this section, we consider weak principal mappings and their some links to modular lattices.

Let be a multiplicative lattice with bottom element and top element . By [2], an element of is called

(i) a weak meet principal element if, for all ,

(ii) a weak join principal element if, for all ,

We now define the corresponding notions for mappings between posets, whose definitions appear more natural than the corresponding ones for elements.

Definition 13. Let be a mapping between posets with as the upper adjoint of .
(i) is called a weak meet principal mapping if
(ii) is called a weak join principal mapping if
If is both a weak meet principal mapping and a weak join principal mapping, then is called a weak principal mapping.

Remark 14. (1) To prove that is a weak meet (weak join) principal mapping, it suffices to show that .
(2) If has an upper adjoint and has a top element ( has a bottom element ), then is weak meet (weak join) principal if and only if = = .

Examples 2. (1) Let be a frame with and as the bottom and top elements, respectively. For any , the mapping , defined by   for all  , is always a weak meet principal mapping. is a weak join principal mapping if and only if there exists such that and   (i.e., has a complement).
(2) Let be a mapping from set to set . Then, the mapping between lattices of power sets of and , respectively, defined by for any , is a weak join principal mapping. is a weak meet principal mapping if and only if is injective.

Proposition 15. Let be a mapping between posets with as the upper adjoint.
(i) If has a top element , then is a weak meet principal mapping if and only if, for all ,
(ii) If has a bottom element , then is a weak join principal mapping if and only if, for all ,

Proof. (i) Suppose is a weak meet principal mapping. Clearly, for any   is a lower bound of . Now, if is a lower bound of , then , implying that for some which implies that . Hence, and so .
Conversely, suppose holds for all . For any , there exists such that . So, . Thus, , so is a weak meet principal.
(ii) follows a dual proof to that of (i).

Using the relation between an element in a multiplicative lattice and the mapping defined by for all , we obtain the following results.

Corollary 16. An element in a multiplicative lattice is weak meet (weak join) principal if and only if the mapping is weak meet (weak join) principal.

Corollary 17. An element in a multiplicative lattice is
(i) weak meet principal if and only if, for any with , there is such that ;
(ii) weak join principal if and only if, for any with , there is such that .

Corollary 17 (i) is Lemma  1 (a) in [2].

Remark 18. If is a meet principal mapping between posets, then holds for all , and so . 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 pentagon , to itself.

Recall that a lattice is said to be modular if, for any with ,    (see, e.g., [12, 13]).

Proposition 19. Let be a mapping between bounded lattices and with an upper adjoint such that is a weak principal mapping.
(i) If is modular, then is a meet principal mapping.
(ii) If is modular, then is a join principal mapping.

Proof. Let be the upper adjoint of .
(i) For any with for some , we have since is weak join principal. Also, which implies that as is weak meet principal. Since , is modular and , As a lower adjoint, preserves joins. Furthermore, which implies that  . Now, Hence, is meet principal.
(ii) follows a proof dual to that of (i).

The following theorem now follows from Proposition 19.

Theorem 20. Let be a mapping between bounded modular lattices with an upper adjoint. Then, is a principal mapping if and only if is a weak principal mapping.

Corollary 21. If is a bounded modular lattice, then every weak principal mapping is principal.

Using Theorem 20, Proposition 9, and Corollary 16, we obtain the following result in [2, 7].

Corollary 22. An element in a modular multiplicative lattice is principal if and only if it is weak principal.

A natural question arising here is whether the converse of Corollary 21 is true. If is a bounded lattice and every weak principal mapping is principal, must   be modular?

Example 23. Consider the lattice where ,  , and . One can check that every weak principal mapping is principal. However,    is not modular.
Another counterexample, suggested by Dr. Peter Jipsen, is the 6-element nonmodular lattice where ,  ,  and  .

The following problem is still open.

Problem 24. Let  be a bounded lattice such that for any bounded lattice , every weak principal mapping is meet principal. Must 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 pentagon , to itself.

Proposition 25. Let and be mappings between posets.
(1) If is weak meet principal and is meet principal, then is weak meet principal.
(2) If is join principal and is weak join principal, then is weak join principal.

Using Propositions 9 and 25 and Corollary 16, we obtain the following result which is Proposition 1  (a) in [2].

Proposition 26. If is a weak meet principal element and is a meet principal element in a multiplicative lattice , then is a weak meet principal element in .

For bounded semilattices, we have another characterization of weak join principal and weak meet principal mappings.

Proposition 27. Let be a mapping between two posets with as the upper adjoint of .
(1) If and are join semilattices with as the bottom element of , then is a weak join principal mapping if and only if, for any  ,   implies that .
(2) If and are meet semilattices with as the top element of , then is a weak meet principal mapping if and only if, for any implies that  .

Proof. As before, we just gave the proof of (1).
Suppose is weak join principal and where . Then, .
Conversely, suppose satisfies the given condition. Clearly, for any . For any , , so, by the assumption on , we have .

By [14], a lower adjoint is injective if and only if its upper adjoint is surjective. Now, if is an injective weak join principal mapping between two join semilattices with bottom elements and ,  respectively, then we must have , where is the upper adjoint of  . Then, implies that , so is an order embedding.

Corollary 28. If is an injective weak join principal mapping between join semilattices with bottom elements, then 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 set , the power set of is a complete lattice with respect to the inclusion order.

Theorem 29. Let and be two nonempty sets. For any mapping , the following statements are equivalent.
(1) is a principal mapping.
(2) There is a subset of and an injective mapping such that for all .

Proof. (1) implies (2). Let be principal and let be its upper adjoint. As every lower adjoint preserves arbitrary joins, it sends bottom element to the bottom element. Thus, .
(i) For any , if is nonempty, then it is a singleton set. As a matter of fact, assume . Then, , where . Since is meet principal, there is a subset such that . Clearly, , so , which implies that is a singleton set.
(ii) Let . Then, because preserves arbitrary joins. It then follows that . Now, by Proposition 15 (ii), for any , . Thus, and , imply that .
(iii) Define by , where . By (ii), is an injective mapping. For any , note that preserves joins (i.e., unions in this case), so .
(2) implies (1). Assume the condition in (2) is satisfied. Then, has an upper adjoint given by for each . For any , if , then for some . Also, which implies that . It follows that . Hence, .
If , then . But is at most a nonempty singleton set. If , then . If , then and , implying that since is injective. It follows that . Therefore, is principal.