Cover Set Lattices

Abstract

The proof of a main result in [1] concerning (0,1)-endomorphisms of finite lattices is based on properties of lattices A(G) derived from the system of independent sets of an undirected loop-free graph G. For a number of questions naturally arising from [1] and [2], however, constructions employing only graph-induced complementation and properties of the lattices A (G) associated with these are no longer adequate. The present paper introduces cover set lattices (a generalization of the lattices A(G)) to deal with some of these questions. A special case of the main result presented here states that for every (0, 1)-lattice L and any monoid homomorphism φ:M → End0,1(L) there exists a lattice K containing L as a (0, 1)-sublattice in such a way that the monoid End0,1(K) of all (0, 1)-endomorphisms of K is isomorphic to M, and the restriction to L of every (0, 1)-endomorphism m of K is the (0, 1)-endomorphism φ(m) of L.