Locally testable semigroups

The locally testable semigroups were discovered in the study of finite automata (see [8], [12] for history and motivation). In this paper, we study the locally testable semigroups from a purely algebraic viewpoint, as a simultaneous generalization of both the normal bands and the nilpotent semigroups. We generalize the results of [8] and [12] to the infinite case and, under mild restrictions on the semigroups, obtain subdirect product decompositions which sharpen these results.