# Cevian operations on distributive lattices

## TL;DR

This paper constructs a special distributive lattice with unique properties that challenge previous assumptions, demonstrating limitations in lattice homomorphisms and setting the stage for further logical and algebraic investigations.

## Contribution

It introduces a new bounded distributive lattice with specific properties that counter previous conjectures and results in lattice theory.

## Key findings

- Constructed a completely normal bounded distributive lattice with unique set properties.
- Showed the lattice does not admit certain binary operations or homomorphic images.
- Established limitations on the class of Stone duals of spectra of Abelian { extless}ell{ extgreater}-groups.

## Abstract

We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x $\in$ D | a $\le$ b $\lor$ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x $\le$ y $\lor$(x-y),(x-y)$\land$(y-x) = 0, and x-z $\le$ (x-y)$\lor$(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex {\ell}-subgroups of any (not necessarily Abelian) {\ell}-group. It has $\aleph 2 elements. This solves negatively a few problems stated by Iberkleid, Mart{\'i}nez, 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 $\lambda$, the class of Stone duals of spectra of all Abelian {\ell}-groups with order-unit is not closed under L $\infty$$\lambda$-elementary equivalence.