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

Larmerová, Jana; Rachůnek, Jiří

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 91, Math. 27, 13-23 (1988).

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)

Keywords:

modular posets; distributive poset; lower homomorphism; translation

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