Lattice conditions implying congruence modularity

Abstract

In J6nsson [6] and Day [1] Mal'cev type characterizations were given for congruence distributivity and congruence modularity respectively. (See e.g. Taylor [9] for the precise definitions and the characterization of Mal'cev type conditions in general.) Gedeonovfi [5] and Mederly [7] showed that other (non-trivial) lattice identities were Mal'cev type conditions but Day [2], Nation [8] and [7] showed that these conditions imply (and hence are equivalent) to either congruence distributivity or congruence modularity. These facts (along with others to be stated below) have raised the following two conjectures (1) McKenzie: If the congruence lattices of a variety satisfy any non-trivial lattice identity, then the variety is congruence modular. (2) Nation: If a non-trivial lattice identity has a Mal'cev type characterization, then it implies congruence modularity. There are particular results in the case of the first conjecture. It has been proven true for varieties of unary algebras (Nation [8]), semigroups (Evans [3] and Freese and Nation [4]) and semilattices ([8]). The first general result, referring to lattice identities of a special form, was due to Nation in [8].