We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.’s). This class seems to be the natural context for studying… (More)

Let A and B be lattices with zero. The classical tensor product, A ⊗ B, of A and B as join-semilattices with zero is a join-semilattice with zero; it is, in general, not a lattice. We define a very… (More)

We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth,… (More)

Say that a cone is a commutative monoid that is in addition conical, i.e., satisfies x+y=0 ⇒ x=y=0. We show that cones (resp. simple cones) of many kinds order-embed or even embed unitarily into… (More)

§0. Introduction. Few methods are known to construct non Lebesgue-measurable sets of reals: most standard ones start from a well-ordering of R, or from the existence of a non-trivial ultrafilter over… (More)

We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results.

In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism Conc(A ⊗ B) ∼ = Conc A ⊗ Conc B, holds, provided that the tensor product satisfies a very natural… (More)

The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Conc L of compact congruences of a… (More)

We prove the following result: Theorem. Every algebraic distributive lattice D with at most א1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results… (More)