Bounded Endomorphisms of Lattices of Finite Height

Abstract

Every monoid M is isomorphic to the monoid End0,1(L) of all (0,1)-preserving endomorphisms of a bounded lattice L, see [3]. The lattices L with M ≌ End0,1(L) that are constructed there are of an arbitrary infinite cardinality not smaller than that of M, and they all have infinite chains. The aim of the present article is to supplement these results. It will be shown that every finite monoid M is representable as End0,1(L) of a finite lattice. In addition, an account of the difficulties involved in attempting to characterize endomorphism monoids of lattices of a fixed finite height will be given.