Monotone-Cevian and finitely separable lattices
Résumé
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 vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean l-group with strong unit, of cardinality $\aleph_1$, whose principal l-ideal lattice does not have a monotone deviation.
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