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 whose principal l-ideal lattice does not have a monotone deviation.
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