Semigroups generated by idempotents arise naturally in many parts of semigroup theory and its applications. In some sense they are as far away from groups as possible and thus require tools of study that are particular to semigroup theory. In this talk, we begin with a brief survey on known results on dempotent generated semigroups. This includes the fact that a semigroup generated by 2 idempotents has at most 6 idempotents but that every (finite) countable semigroup embeds into a (finite) semigroup generated by 3 idempotents.
Nambooripad developed a deep theory of biordered sets in order to study idempotents of semigroups with products restricted to so called basic products- those coming from a pair of idempotents related by one of Green's order relations R and L. He also developed a theory of inductive groupoids and established a category equivalence between these and regular semigroups.
As part of this theory, he showed that every regular biordered set E gives rise to a freest idempotent generated semigroup IG(E) whose biordered set is precisely E. He constructed the inductive groupoid of IG(E) directly from E, as a quotient of the groupoid of E-chains, where an E-chain is a product of idempotents where each pair of neighboring factors are related by one of Green's equivalence relations R or L.
The main purpose of this talk is to study the maximal subgroups of IG(E). We do this topologically by identifying each maximal subgroup as the fundamental group of a squared-2-complex (each two cell has a border of length 4) whose underlying graph is the so called Graham-Houghton graph, a tool that was developed in the late 1960's to study idempotent generated subsemigroups of finite semigroups.
We give the first examples of biordered sets E such that IG(E) has maximal subgroups that are not free. One comes from the biordered set of 3 x 3 matrices over the field of order p and one comes from defining a biordered set over a squareification (like triangularization!) of the surface of the torus S1 x S1.
Although the theory of biordered sets and inductive groupoids is quite technical, the speaker will do his best to keep the talk understandable to people with a background in semigroup through Green's relations and the Rees Theorem.
This is joint work with John Meakin.