Congruences on ωⁿ-Bisimple Semigroups

Abstract

Let S be a bisimple semigroup and let E s denote its set of idempotents. We may partially order E s in the following manner: if e , f ∈ E s , e ≧ f if and only if ef = fe = e . We then say that E s is under or assumes its natural order. Let I 0 denote the non-negative integers and let n denote a natural number. If E s , under its natural order, isomorphic to ( I 0 ) n under the reverse of the usual lexicographic order, we call S an ω n -bisimple semigroup. (See [9] for an explanation of notation.) We determined the structure of ω n -bisimple semigroups completely mod groups in [9]. The ω n -bisimple semigroups, the I -bisimple semigroups [8], and the ω n I -bisimple semigroups [9] are classes of simple semigroups except completely simple semigroups whose structure has been determined mod groups.