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 condition that fails in the variety of semilattices. (c) The rectangular commutator is trivial throughout V . (II) (a) V satisfies a nontrivial meet continuous congruence identity. (b) V satisfies an idempotent Maltsev condition that fails in the variety of sets. (c) The strong commutator is trivial throughout V . (d) The strong rectangular commutator is trivial throughout V . (III) (a) V is congruence semidistributive. (b) V satisfies an idempotent Maltsev condition that fails in the variety of semilattices and in any nontrivial variety of modules. (c) The rectangular and TC commutators are both trivial throughout V . We prove that a residually small variety that satisfies a congruence identity is congruence modular. 1991 Mathematics Subject Classification. Primary 08B05; Secondary 08B10.