Congruence semimodular varieties I: Locally finite varieties

The lattice of closed subsets of a set under such a closure operator is semimodular. Perhaps the best known example of a closure operator satisfying the exchange principle is the closure operator on a vector space W where for X ___ W we let C(X) equal the span of X. The lattice of C-closed subsets of W is isomorphic to Con(W) in a natural way; indeed, if Y _~ W x W and Cg(Y) denotes the congruence on W generated by Y, then the closure operator Cg satisfies the exchange principle. For another example, let C denote the closure operator on the set A x A where C(X) equals the equivalence relation generated by X ~_ A 2. This closure operator satisfies the exchange principle, so the lattice of all equivalence relations on A is semimodular. Equivalently, the congruence lattice of any set is semimodular. The preceding examples suggest to us that semimodularity may be a natural congruence condition worth investigating. Research into varieties of algebras with modular congruence lattices had led to the development of a deep structure theory for them. We wonder: how much of the structure involved in congruence modular varieties exists for congruence semimodular varieties? How much more diversity is permitted? This paper may be considered to be an attack on the former question