Cevian operations on distributive lattices

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 $\vee$ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x $\le$ y $\vee$(x-y),(x-y)$\wedge$(y-x) = 0, and x-z $\le$ (x-y)$\vee$(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 $\aleph 2elements.ThissolvesnegativelyafewproblemsstatedbyIberkleid,Mart\acute{ı}nez,andMcGovernin2011andrecentlybytheauthor.Thisworkalsoservesaspreparationforaforthcomingpaperinwhichweprovethatforanyinfinitecardinal$λ$,theclassofStonedualsofspectraofallAbelian\ell -groupswithorder-unitisnotclosedunderL$\infty$$λ$-elementary equivalence.

23 pages. v2 removes a redundancy from the definition of a Cevian operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on the G side)