Monotone-Cevian and Finitely Separable Lattices

Abstract

AbstractA 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 ) ↦ x \ y satisfying the rules $$x\le y\vee (x\mathbin {\smallsetminus }y)$$ x ≤ y ∨ ( x \ y ) and $$(x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0$$ ( x \ y ) ∧ ( 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., $$x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)$$ x \ z ≤ ( x \ y ) ∨ ( 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 $$\ell $$ ℓ -ideals of Abelian $$\ell $$ ℓ -groups (which are always completely normal). We prove that for free Abelian $$\ell $$ ℓ -groups (and also free $$\Bbbk $$ k -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean $$\ell $$ ℓ -group with strong unit, of cardinality $$\aleph _1$$ ℵ 1 , whose principal $$\ell $$ ℓ -ideal lattice does not have a monotone deviation.