The homomorphicity of certain maps that appear in studying the lattices of processes in graphs

Abstract

Necessary and sufficient conditions for the homomorphicity of certain maps pertaining to the lattices of processes in graphs are presented. The results are stated in a graph-independent form. Bibliography: 2 titles.In the course of studying the lattices in graphs [1], the question concerning the homomorphicity of certain maps arises.The present paper provides necessary and sufficient conditions for the homomorphicity of these maps. The conditions are stated in a graph-independent form.Below we use the following notation: L is a distributive lattice; e, e I are arbitrary elements of L; DL is the lattice of filters of L; IL is the lattice of ideals of L; f~, F, are mappings determined by e' ---, e' A e, e' ---+ e' V e, which map L onto Proof. First we note that17^ 2 1-I1 u II~.Indeed, because of K^ < K1, K2 and the fact that 5 r [1] is antiisotone, we have H^ D I-I1, l-i^ _D II2.The relation a E [S) is equivalent to the existence of a finite .~I,,, M,, C S: AM,, _< a.