Monotone-Cevian and Finitely Separable Lattices

Abstract 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) x y (x, y) ↦ x \ y satisfying the rules x y (x y) x ≤ y ∨ (x \ y) and (x y) (y 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 z (x y) (y 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 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 ₁ ℵ 1, whose principal ℓ -ideal lattice does not have a monotone deviation.