  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.
