Modularity in topological lattices
Don E. Edmondson · Proceedings of the American Mathematical Society · 1969
The purpose of this note is to establish that for topological lattices of suitably small breadth, connectedness implies modularity without an exploitation of compactness. L will denote a topological lattice, that is a Hausdorff topological space with continuous binary operations V and A for which (L, V, A) is a lattice. For a more explicit presentation and some related machinery, see the paper of E. Dyer and A. Shields [5]. Of specific need, L is a modular lattice iff for every a, b, cCL, b<a implies aA(bVc)=bV(aAc). And if n is a positive integer, the breadth of L is less than n means that for every xi, X2, * * EzCL, there exists j such that x < Vioxi. Note that the breadth formulation given here is dual to the equivalent phrasing used in [5]. Also for a<b in L, define [a, b] = {xELI a<x?b }.