Monotone-Cevian and Finitely Separable Lattices

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) bar right arrow x \ y satisfying the rules x = y boolean OR (x \ y) and (x \ y) boolean AND (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 <= (x \ y) boolean OR (y \ z)). We relate thosematters 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 k-vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean l-group with strong unit, of cardinality N-1, whose principal l-ideal lattice does not have a monotone deviation.