Semihomomorphisms of semimodular lattices

Abstract

In this paper a class of lattice is defined in a manner analogous to that of lattice semimodularity. It is shown that a semihomomorphism of a modular lattice is a homomorphism; while on a semimodular lattice which satisfies a certain regularity property, semihomomorphisms are uniquely determined by specifying their kernels. The definitions and results are given below for the lower semimodular case; their duals for upper semimodularity are straight forward. With the exception of Theorems 3 and 12, all results extend immediately to the case where ? is a finite-dimensional lower semilattice, and this fact is used in the proof of Theorem 12. For any element x H0 of a lattice ?, let Ax denote the set of atoms a C ? (elements which cover 0) such that a 2. A lattice is said to be lower semimodular (LSM) if xVy covers both x and y, then x and y both cover xAy. It is well known that semimodular lattices satisfy the Jordan-Dedekind condition and that dimension is additive [1]. A-regular, lower semimodular lattices will be called G-lattices. G-lattices arise naturally in the study of directed graphs; if one partially orders by inclusion the set of convex subgraphs of any finite directed graph, then the resulting lattice is lower semimodular and A-regular [2]. G-lattices have several interesting structural properties, but we shall need only the result given below.