TITLE: examples of finitely generated semigroups that are not residually finite
QUESTION [5 upvotes]: Does anybody know of any finitely generated semigroups that are not residually finite and whose group of units (if there is an identity) is trivial?  Basically, I'm looking for finitely generated semigroups that are not residually finite, but I don't just want to punt and look at groups (like Thompson's groups or something).  Thanks!

REPLY [3 votes]: To generalise @BenjaminSteinberg's (by now rather old) answer, Lallement showed in 1971 that a one-relation monoid $M = \text{Mon}\langle A \mid w= w'\rangle$ where $w$ is a left and right factor of $w'$ (that is, there exist $w_1, w_2 \in A^\ast$ with $w' \equiv ww_1 \equiv w_2 w$, where $\equiv$ denotes graphical equality of words) is not residually finite if the presentation relation of the left special monoid $L(M)$ associated to $M$ is not the presentation relation of a group. In this case, it cannot be embedded into a compact semigroup, just as the bicyclic monoid.
The left special monoid comes about through so-called compression, which is a little too long to get into here, but one particular corollary is that if one takes $w \equiv \varepsilon$, then $L(M) = M$. Hence $\text{Mon}\langle A \mid w' = 1\rangle$ is not residually finite if it is not the free product of a free monoid by a group. So this includes the bicyclic monoid $\text{Mon}\langle b,c \mid bc = 1\rangle$, but also more general examples such as $\text{Mon}\langle a,b \mid ababb = 1\rangle$ and $\text{Mon}\langle a,b,c \mid abacac = 1\rangle$, both of which have trivial group of units.
One can  easily check whether or not a one-relation special monoid $\text{Mon}\langle A \mid w' = 1\rangle$ is the free product of a free monoid by a group by Adian's algorithm.