Idempotent Simple Algebras*

Abstract

We show that every idempotent simple algebra which has a skew congruence in a power either has an absorbing element or is a subreduct of an affine module. We refine this result for idempotent algebras which have no nontrivial proper subalgebras. One corollary we obtain is a new proof of Á. Szendrei's classification theorem for minimal locally finite idempotent varieties. We partially extend this classification to nonlocally finite varieties. Another corollary is a complete classification of all minimal varieties of modes.