## Translations of distributive and modular ordered sets.(English)Zbl 0693.06003

The authors define distributive and modular posets in a manner that generalizes the corresponding notions from lattice theory. Both notions turn out to be selfdual, and it is shown that every distributive poset is modular. For a poset P, let U(A), L(A) denote the set of upper, lower bounds of the subset A of P. Distributivity is defined by the requirement that $$L(U(a,b),c)=L(U(L(a,c),L(b,c))),$$ and modularity by $$U(L(a,b),c)=U(L(U(a,c),U(b,c)))$$ whenever $$a\leq c$$. The mapping f on P is called a lower homomorphism if $$U(f(L(a,b)))=U(L(f(a),f(b)))$$ for all a, b in P, and it is called a translation if $$f(U(a,b))=U(f(a),b)$$. The connection between lower homomorphisms and translations is explored when P is a distributive or modular poset. It is also established that when P is a lattice, then lower homomorphisms coincide with meet homomorphisms.
Reviewer: M.F.Janowitz

### MSC:

 06A06 Partial orders, general 06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text:

### References:

 [1] Rachůnek J.: Translations des ensembles ordonnés. Math. Slovaca 31 (1981), 337-340. · Zbl 0472.06002 [2] Szàsz G.: Translationen der Verbände. Acta Univ. Comen., Math., 5 (1961), 449-453. · Zbl 0112.01901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.