# Cevian operations on distributive lattices

- Authors
- Publication Date
- Feb 17, 2019
- Source
- HAL-UPMC
- Keywords
- Language
- English
- License
- Unknown
- External links

## Abstract

We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x ∈ D | a ≤ b ∨ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x ≤ y ∨(x-y),(x-y)∧(y-x) = 0, and x-z ≤ (x-y)∨(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex ℓ-subgroups of any (not necessarily Abelian) ℓ-group. It has ℵ 2 elements. This solves negatively a few problems stated by Iberkleid, Martínez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal λ, the class of Stone duals of spectra of all Abelian ℓ-groups with order-unit is not closed under L ∞λ-elementary equivalence.