Principal Mappings between Posets

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 R, 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.


Introduction
A multiplicative lattice [1][2][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 1  as the multiplication identity.
The complete lattice Idl() 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 ,  ∈ Idl() where [ : ] is the ideal quotient [4].
In terms of the order and the multiplication on the lattice Idl(), 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][6][7][8][9][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   ,   : Idl() → Idl() by for each  ∈ Idl(). 2 International Journal of Mathematics and Mathematical Sciences By (2), if  is a principal ideal of , then   and   satisfy the following equations for any ,  ∈ Idl(): ( ∧   ()) =   () ∧ ,   ( ∨   ()) =   () ∨ . (4) Thus, every principal ideal  of  corresponds to a special mapping   from the lattice Idl() 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 Idl() 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.

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 .
(3) Let (Idl(), ⊆) be the lattice of ideals of a commutative ring  with inclusion as the partial order.For any principal ideal  0 ∈ Idl(), the mapping   0 : Idl() → Idl() is a principal mapping where   0 () =  0  (the multiplication of  0 with ) for any  ∈ Idl().The residual of   0 sends  to the ideal quotient [ :  0 ].See Theorem 7 for the general result on lattices.
(4) Let  be a topological space and O() be the complete lattice of open sets of .Given any  ∈ O(), define the mapping   : O() → O() by   () =  ∩ ,  ∈ O().Then,   is a meet principal mapping that is not a join principal mapping unless  is closed.
( 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 .
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 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.
In the case where  and  are semilattices or lattices, we have a neater characterization of the types of mappings defined in Definition 1.
The following proposition easily follows from the definition of principal mappings.
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 𝑎 of a multiplicative lattice 𝐿 is a meet (join) principal element if and only if the mapping
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  = [ : ].

International Journal of Mathematics and Mathematical Sciences
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 PRMP(, ) be the set of all principal mappings from the multiplicative lattice  to itself, and let PRE() be the set of all principal elements of .Then, (PRMP(, ), ∘) (∘ denotes the composition operation) is a semigroup with id  as the identity, and (PRE(), ⋅) is also a semigroup with 1  as the identity, and ⋅ as the multiplication on .Now the mapping   →   defines an embedding of (PRE(), ⋅) into (PRMP(, ), ∘).
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, Idl() 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  ∈ ?

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 0  and top element 1  .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 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 ↓ () ⊆ ()(↑ () ⊆ (), resp.).
(2) If  :  →  has an upper adjoint  :  →  and  has a top element 1  ( has a bottom element 0  ), then  is weak meet (weak join) principal if and only if Examples 2. (1) Let  be a frame with 0  and 1  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  ∧  ⊥ = 0  and  ∨  ⊥ = 1  (i.e.,  has a complement).
(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 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  5 , to itself.
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.
(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.

Corollary 22. An element in a modular multiplicative lattice 𝐿 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?

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  5 , to itself.Proposition 25.Let  1 :  1 →  2 and  2 :  2 →  3 be mappings between posets.
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 0  as the bottom element of , then  is a weak join principal mapping if and only if, for any ,  ∈ , () ≤ () implies that  ≤ ∨(0  ).
(2) If  and  are meet semilattices with 1  as the top element of , then  is a weak meet principal mapping if and only if, for any ,  ∈ , () ≥ () implies that  ≥  ∧ (1  ).
Proof.As before, we just gave the proof of (1).
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 0  and 0  , respectively, then we must have (0  ) = 0  , where  :  →  is the upper adjoint of .Then, () ≤ () implies that  ≤  ∨ (0  ) =  ∨ 0  = , 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.

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 P() of  is a complete lattice with respect to the inclusion order.
Theorem 29.Let  and  be two nonempty sets.For any mapping  : P() → P(), the following statements are equivalent.
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, (0) = 0.
For contractive principal mappings on lattices of powersets, we have the following.
Proposition 32.A mapping  : P() → P() is a contractive principal mapping if and only if there exists a subset  1 of  such that () =  ∩  1 for all  ∈ P().
Problem 33.Determine the complete lattices  such that for any contractive principal mapping  :  → , there is an element  ∈  such that () =  ∧  for all  ∈ .
Recall that a chain is a poset in which every two elements are comparable.
Theorem 34.Let  be a chain and let  :  →  be a mapping.Then, the following statements are equivalent.
If  does not have a bottom element, then ( 2) is equivalent to the following.
(3)  is a meet principal mapping that is strictly monotone.
Corollary 35.For any chain  with  elements, there are exactly  principal mappings  :  → .
One of the main tasks we are interested in is to determine all the principal mappings  : Idl() → Idl() where Idl() is the lattice of ideals of a commutative ring .
Proof.(Z   , +, ⋅) is a principal ideal domain and there are exactly  + 1 ideals forming a chain.So the first part follows from Corollary 35.For the second part, just note that each   ( ∈ Idl(Z   )) is principal and that  ̸ =  for  and  ∈ Idl(Z   ) implies that   ̸ =   .
An element  of a complete lattice  is called a pseudoprincipal element if there is a contractive principal mapping  :  →  such that (1  ) = .
If  is a principal element of a multiplicative lattice , then  =   (1  ), so  is pseudoprincipal.The proof of Corollary 35 also shows that every element in a finite chain is pseudoprincipal.

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 ideal  of a commutative ring , the mapping   : Idl() → Idl() given by   () =  for all  ∈ Idl() will be called the mapping defined by .Note that   is always a lower adjoint of the mapping   : Idl() → Idl() defined by   () = [ : ] for any  ∈ Idl().By Proposition 9 (ii),   is a principal mapping if and only if  is a principal element of the multiplicative lattice Idl().
Definition 37. A multiplicative lattice  has the divisibility order if, for all ,  ∈  and  ≤  if and only if  divides  (i.e.,  =  for some  ∈ ).
Recall that a cancellation ideal ring [15] is a commutative ring  in which every nonzero ideal  of  is a cancellation ideal [16]; that is, whenever  =  for any ,  ∈ Idl(), and then  = .It is easy to see that the above condition that is equivalent to  ⊆  implies that  ⊆ .
Note that every principal ideal domain  is a cancellation ideal ring and the corresponding multiplicative lattice Idl() has the divisibility order.
Theorem 38.Let  be a cancellation ideal ring such that Idl() has the divisibility order.Then,  : Idl() → Idl() is an injective principal mapping if and only if there is an order isomorphism  : Idl() → Idl() and a nonzero principal element  of Idl() such that  =   ∘ .
Proof.Only the necessity needs verification because every isomorphism is a principal mapping and the composition of principal mappings is principal.
In the following, for any element  of a commutative ring, ⟨⟩ denotes the principal ideal generated by .
The following fact will be used several times in later arguments.
Lemma 39.Let  be a principal ideal domain.For any nonzero  ∈ , any prime element  in  and  ∈ Z + ⟨  ⟩ ⊆ ⟨⟩ if and only if  =   for some unit  and some 0 ≤  ≤ .
Recall that an element  of a lattice  is called a (meet) irreducible element if, for any , V ∈ ,  =  ∧ V implies that  =  or  = V.Lemma 40.Let  be a principal ideal domain.Then, ⟨  ⟩, where  is a prime element in  and  ∈ Z + , are exactly the irreducible elements of the lattice Idl().
Conversely, let ⟨⟩ be an irreducible element of the lattice Idl() and let  = be a prime factorization of  (note that, in any UFD, in particular, a PID, prime elements coincide with (ring) irreducible elements), where  is a unit and   are distinct prime elements of .Then, ⟨⟩ = ⟨ Lemma 41.Let  be a principal ideal domain.If  : Idl() → Idl() is an order isomorphism, then, for any prime element  of , there is a prime element  of  such that for any  ∈ Z + , (⟨  ⟩) = ⟨  ⟩.
We are now able to prove the main result of this section.
Theorem 42.Let  be a principal ideal domain.Then,  : Idl() → Idl() is an injective contractive principal mapping if and only if  =   for some nonzero  ∈ Idl().
Proof.By Theorem 38, there exist an order isomorphism  : Idl() → Idl() and a nonzero ideal  of  such that  =   ∘  where   is the mapping defined by .

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 ring , determine all the principal mappings from the lattice Idl() to itself.
(2) Determine, for which ring , every injective contractive principal mapping from Idl() to itself is defined by an ideal of .
Let Z[] be the polynomial ring over the ring of integers.Is there an injective contractive principal mapping  : Idl(Z[]) → Idl(Z[]) that is not defined by any ideal of Z[]?
) Let  :  →  be a continuous open mapping between topological spaces  and  (i.e., () is an open set of  for any open set  of ).Let  * : O() → O() be defined by  * () = (),  ∈ O().Then,  * is a meet principal mapping.In general,  * is not a join principal mapping.

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.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  ≥ [0 : ], there is  ∈  such that  = [ : ].