Monotone-Cevian and Finitely Separable Lattices

Abstract

A distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation \((x,y)\mapsto x\mathbin {\smallsetminus }y\) satisfying the rules \(x\le y\vee (x\mathbin {\smallsetminus }y)\) and \((x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0\) — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., \(x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)\)). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal \(\ell \)-ideals of Abelian \(\ell \)-groups (which are always completely normal). We prove that for free Abelian \(\ell \)-groups (and also free \(\Bbbk \)-vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean \(\ell \)-group with strong unit, of cardinality \(\aleph _1\), whose principal \(\ell \)-ideal lattice does not have a monotone deviation.