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 {\ell}-subgroups of any (not
necessarily Abelian) {\ell}-group. It has ℵ2elements.ThissolvesnegativelyafewproblemsstatedbyIberkleid,Martiˊnez,andMcGovernin2011andrecentlybytheauthor.Thisworkalsoservesaspreparationforaforthcomingpaperinwhichweprovethatforanyinfinitecardinal\lambda,theclassofStonedualsofspectraofallAbelianℓ−groupswithorder−unitisnotclosedunderL\infty\lambda$-elementary equivalence.Comment: 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