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-y satisfying the rules x \leq y\vee (x-y) and (x-y) \wedge (y-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-z \leq (x-y) \vee (y-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 l-ideals of Abelian l-groups (which are always completely normal).We prove that for free Abelian l-groups (and also free vectorlattices) those lattices admit monotone Cevian deviations.On the other hand, we construct an Archimedean l-group with strong unit whose principal l-ideal lattice does not have a monotone deviation.