Congruences on ωn-Bisimple Semigroups

Abstract

Let S be a bisimple semigroup and let Es denote its set of idempotents. We may partially order Es in the following manner: if e, f ∈ E s, e ≧ f if and only if ef = fe = e. We then say that Es is under or assumes its natural order. Let I0 denote the non-negative integers and let n denote a natural number. If Es, under its natural order, isomorphic to (I0)n under the reverse of the usual lexicographic order, we call S an ωn-bisimple semigroup. (See [9] for an explanation of notation.) We determined the structure of ωn-bisimple semigroups completely mod groups in [9]. The ωn-bisimple semigroups, the I-bisimple semigroups [8], and the ωnI-bisimple semigroups [9] are classes of simple semigroups except completely simple semigroups whose structure has been determined mod groups.