Homotopy Types of Small Semigroups

We algorithmically compute integral Eilenberg-MacLane homology of all semigroups of order at most $8$ and present some particular semigroups with notable classifying spaces, refuting conjectures of Nico. Along the way, we give an alternative topological proof of the fact that if a finite semigroup $S$ has a left-simple or right-simple minimal ideal $K(S)$, then the classifying space $BS$ is homotopy equivalent to the classifying space $B(GS)$ of the group completion. We also describe an algorithm for computing the group completion $GS$ of a finite semigroup $S$ using asymptotically fewer than $\mathrm{\mid }S{\mathrm{\mid }}^2$ semigroup operations. Finally, we show that the set of homotopy types of classifying spaces of finite monoids is closed under suspension.