Let N denote the free semigroup of rank 1. Say that a semigroup T is a small extension of N if N embeds in T and T  N is finite. Is there some kind of classification of small extensions of N? What do these semigroups look like?

David, I don't know if you still care about this. The people at St. Andrews say that a semigroup S has finite Rees index in a semigroup T if S is a subsemigroup of T and TS is finite. They prove that this forces S and T to have a lot in common. For example, if memory serves one is finitely presented iff the other is. Probably the same is true for things like residual finiteness. I don't know if their results help for what you want. 


This is probably not the kind of answer you are looking for, but we can take any finite semigroup $R$, and define $T=N\sqcup R \sqcup \{\theta\}$ where $\theta$ is a formal symbol; we then make $T$ into a semigroup by declaring the product of $\theta$ with anything to be $\theta$ itself, declaring that $nr=\theta=rn$ for all $n\in N$ and all $r\in R$, and then keeping the same products on $N$ and $R$ as before. Then $T$ is a semigroup, $N$ embeds as a subsemigroup, and $N$ is a cofinite subset of $T$; but I find it hard to see how we can say anything useful about $T$. Basically, I think you need to come up with a refined version of your question, perhaps by restricting $T$ to lie in some class of semigroups which you care about. Update: In the comments, David asks if a finite extension of N (in the sense defined in the question) must contain a left or right zero for example. It is not too difficult to see that this need not be the case; take a finite semigroup S (which necessarily has a zero element) and stick it at the front of N, so that the product of $s\in S$ and $n\in N$ is just $n$. 

