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 \((x,y)\mapsto x\mathbin {\smallsetminus }y\) $(x,y)\mapsto x\backslash y$ satisfying the rules \(x\le y\vee (x\mathbin {\smallsetminus }y)\) $x\le y\vee (x\backslash y)$ and \((x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0\) $(x\backslash y)\wedge (y\backslash x)=0$ — 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., \(x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)\) $x\backslash z\le (x\backslash y)\vee (y\backslash z)$ ). 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 \(\ell \) $\ell$ -ideals of Abelian \(\ell \) $\ell$ -groups (which are always completely normal). We prove that for free Abelian \(\ell \) $\ell$ -groups (and also free \(\Bbbk \) $\mathbb{k}$ -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean \(\ell \) $\ell$ -group with strong unit, of cardinality  \(\aleph _1\) ${\aleph }_1$ , whose principal \(\ell \) $\ell$ -ideal lattice does not have a monotone deviation.