On distributive and modular \(\chi\)-lattices. (English) Zbl 0532.06002

For two elements a, b of a partially ordered set P let \(mub\{a,b\}\) and \(mlb\{a,b\}\) denote the (eventually empty) set of the minimal upper resp. the maximal lower bounds of the two-element set \(\{\) a,\(b\}\). In this paper the author considers the particular case when \(mub\{a,b\}\) and \(mlb\{a,b\}\) are nonempty for every pair \(a,b\in P.\) Let \(\chi\) be a function choosing for each \(a,b\in P\) a single element from \(mub\{a,b\}\) as well as from \(mlb\{a,b\}.\) The algebra \((P,\vee,\wedge)\) defined by \(a\vee b=\chi(mub\{a,b\})\) and \(a\wedge b=\chi(mlb\{a,b\})\) is called a \(\chi\)-lattice derived from P. It is easy to see that these operations are commutative and satisfy the absorption laws but they are non- associative in general. Distributivity and modularity of \(\chi\)-lattices are defined by means of the so-called distributive resp. modular inequalities. It is shown that distributivity implies modularity, moreover the classical Birkhoff condition for distributivity resp. the Dedekind condition for modularity (by subsets) is proved to be valid for \(\chi\)-lattices, too. Conditions for a \(\chi\)-lattice to be a join- or meet-semilattice are given. Translations and congruences on \(\chi\)- lattices are defined in the same manner as on lattices and their basic properties are shown to be unaltered.
Reviewer: G.Szász


06A06 Partial orders, general
06B99 Lattices