Elsevier

Information Sciences

Volume 180, Issue 6, 15 March 2010, Pages 971-983
Information Sciences

On weakly cut-stable maps

https://doi.org/10.1016/j.ins.2009.11.025Get rights and content

Abstract

A mapping f:P→Q between posets P and Q is called weakly cut-stable if f can be naturally extended to a weakly complete lattice homomorphism (i.e., preserving non-empty meets and joins) f∗:N(P)→N(Q) between the corresponding Dedekind–MacNeille completions. These mappings are studied and described by the 1st-order language.

Section snippets

Preliminaries

It is well known that algebraic structures, i.e. sets with operations and relations, can capture functioning of many real systems. This fact is of great importance from the point of view of applications, for example in the theory of switching circuits and the theory of automata.

Recall that the notion of automaton appeared at the junction of several branches of science, namely engineering, biology and computer science. It can be considered as a system with input and output channels. The

Weakly cut-stable maps

Definition 2.1

A mapping f:P→Q will be called

  • (a)

    weakly lower cut-stable (WLCS) if

    • 1.

      L(f(U(A)))=LU(f(A)) whenever ∅≠A⊆P,U(A)≠∅,

    • 2.

      U(f(A))=U(f(P)) for ∅≠A⊆P with U(A)=∅;

  • (b)

    weakly upper cut-stable (WUCS) if

    • 3.

      U(f(L(A)))=UL(f(A)) whenever ∅≠A⊆P,L(A)≠∅,

    • 4.

      L(f(A))=L(f(P)) for ∅≠A⊆P with L(A)=∅.

If f is both WLCS and WUCS, then f will be called weakly cut-stable (WCS).

By a weakly complete lattice homomorphism we mean a homomorphism between complete lattices preserving both non-empty meets and non-empty joins.

The crutial result of this

Properties of weakly cut-stable maps

Definition 3.1

A mapping f:P→Q will be called weakly cut-preserving (WCP) if it satisfies the condition: if (A,B) is a cut in P with A,B≠∅, then (L(f(B)),U(f(A))) is a cut in Q.

Evidently, the fact that (L(f(B)),U(f(A))) is a cut can be expressed as LU(f(A))=L(f(B)), or equivalently, as UL(f(B))=U(f(A)).

Lemma 3.2

Every WLCS (WUCS) map is WCP.

Proof

Let f:P→Q be WLCS and let (A,B) be a cut in P with A,B≠∅. Then U(A)=B,L(B)=A, so that L(f(B))=L(f(U(A)))=LU(f(A)) by 1. □

Recall that a mapping f:P→Q is isotone if f(x)⩽f(y) whenever x

Characterizing WCS maps by the 1st-order language

As one can immediately see from the definition of weakly cut-stable maps, the properties 1–4 are not given in the 1st-order language. Hence to verify that a given mapping f:P→Q is WCS may be rather time consuming when P is taken “rather big”. In what follows we describe WCS, WLCC and WUCC maps in the 1st-order language, i.e. we characterize them by formulas in which quantifiers range over one-element subsets.

Theorem 4.1

A mapping f:P→Q is WLCS iff f is ISO and for all g,r∈Q,g≰r,f-1(L(r))≠∅ the following

Conclusion

Since algebraic structures connected with many real systems bear an order relation, we were interested in the general question which mappings between partially ordered sets can be extended in a natural way to complete (but not 0-1) homomorphisms of the corresponding Dedekind–MacNeille completions. Our result is that these mappings are exactly so-called weakly cut-stable mappings. Moreover, we described weakly cut-stable mappings by the first-order language.

Acknowledgement

This work was supported by Science and Technology Assistance Agency under the contract No. APVV-0007-07, by the Grant VEGA 1/3003/06 and by the Research and Development Council of Czech Government via the project MSN 6198959214.

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