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)↦ 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.