Abstract. The equivalences ${\cal D}, {\cal L}$ and ${\cal R},$ defined initially on semigroups by J. A. Green, are used to study both noncommutative lattices and their congruence lattices, with particular attention given to the effects of assuming that some or all of these equivalences are congruences. Several specialized classes of noncommutative lattices are considered, including some that are simple algebras. Occurrences of distribution in noncommutative lattices as well as their congruence lattices are also considered.

AMS Subject Classification (1991): 06F05, 08A30, 20M10

Keyword(s): noncommutative lattices, congruences, Green's equivalences

Received February 7, 2001, and in revised form January 26, 2002. (Registered under 2854/2009.)