SOME ELEMENTARY PROPERTIES OF CONDITIONALLY DISTRIBUTIVE LATTICES
Jacek Hawranek, Jan Zygmunt · 1983
The notion of a conditionally distributive lattice was introduced by B. Wolniewicz while formally investigating the ontology of situations (cf. [2]). In several of this lectures he has appealed for a study of that class of lattices. The present abstract is a response to that request. 1. A characterization theorem We shall consider only distributive lattices which have unit elements. Definition 1. A lattice L (with the unit element 1) is conditionally distributive, in short, c-distributive, iff the following clause holds: (c) b+ c 6 = 1 ⇒ a(b+ c) = ab+ ac for all a, b, c ∈ L. It is easily seen that the class of c-distributive lattices is closed under homomorphisms and sublattices, but it is not closed under direct products. The two typical examples of non-c-distributive lattices areM⊕5 and N 5 whose diagrams are as follows: