Coclass theory for nilpotent semigroups

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. In joint work with Bettina Eick I investigate a similar approach for the study of finite nilpotent semigroups. This differs from the group theorectic setting in that we additionally use certain algebras associated with the semigroups. A semigroup $S$ is nilpotent if the set $S^c$ of all products of length $c$ has size 1 for some natural number $c$. The smallest such number is called the nilpotency class of $S$, and $|S|-c$ is called the coclass. Given a field $K$ we associate with $S$ the contracted semigroup algebra in which the zeros of $S$ and $K$ are identified and the remaining elements of $S$ form a basis. We visualise the isomorphism types of nilpotent semigroups of coclass $r$ using a graph $G_{r,K}$. The vertices of $G_{r,K}$ correspond to the isomorphism types of contracted semigroup algebras of semigroups of coclass $r$. Two vertices $A$ and $B$ are adjoined by a directed edge from $A$ to $B$ if the quotient $B/B^{c-1}$ is isomorphic to $A$, where $c$ is the class of any semigroup generating $B$. We investigated various graphs $G_{r,K}$ and observed that they share the same general features. I will report on the investigation and describe a number of conjectures that we drew from our observations.