Lattices in Z2and the congruence xy+uv ≡c (mod m)

Algebraic lattices as congruence lattices: The m-complete case

Congruence Lattices of Uniform Lattices

A lattice L is uniform, if for any congruence Θ of L, any two congruence classes A and B of Θ are of the same size, that is, |A| = |B| holds. A classical result of R. P. Dilworth represents a finite distributive lattice D as the congruence lattice of a finite lattice L. We show that this L can be constructed as a finite uniform lattice.

The Shape of Congruence Lattices

We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras. (I) (a) V satisfies a nontrivial congruence identity. (b) V satisfies an idempotent Maltsev cond...

Distributive Congruence Lattices of Congruence-permutable Algebras

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

Congruence Lattices of Congruence Semidistributive Algebras

Nearly twenty years ago, two of the authors wrote a paper on congruence lattices of semilattices [9]. The problem of finding a really useful characterization of congruence lattices of finite semilattices seemed too hard for us, so we went on to other things. Thus when Steve Seif asked one of us at the October 1990 meeting of the AMS in Amherst what we had learned in the meantime, the answer was...