The fundamental four-spiral semigroup

Abstract

Pastijn [17] showed that every semigroup may be embedded in a bisimple idempotent-generated semigroup. This resolves in the negative the conjecture of Eberhart, Williams, and Kinch [4] that a simple idempotent-generated semigroup is completely simple and suggests a study of the structure of bisimple idempotent-generated semigroups which are not completely simple. In this paper we introduce and analyse the structure of a semigroup which is a basic building block for bisimple non-completely simple idempotent-generated semigroups. This semigroup is fundamental (in the sense of Munn [13] and Nambooripad [15]) and has a biordered set of idempotents which can be described as a “spiral” biordered set: It is idempotent-generated, generated by four idempotents and can be described as a rectangular band of four semigroups, three of which are isomorphic to the bicyclic semigroup and one of which is a union of a bicyclic semigroup and an infinite cyclic semigroup.