Semigroup with exactly one left(right) identity?

Question

Are there any examples of a semigroup (which is not a group) with exactly one left(right) identity (which is not the two-sided identity)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects) or they could only be “manually constructed” from abstract symbols (a, b, c…) subject to operation given by a Caley table?

Answer 1

Take a finite semigroup $S$. Then $S$ has an idempotent element $e$ since $S$ is finite.

Let $T = \{se : s \in S\}$. Then $T$ is a subsemigroup of $S$. We have $e \in T$ because $e = ee$. And $e$ is a right identity of $T$ since $(se)e = s(ee) = se$ for all $s \in S$.

My problem with this example is that I don't think $e$ is the only right identity of $T$ for every such $T$ and this is probably still not real world enough.

I believe if we give more conditions when we construct $T$, we might be able to get a semigroup with only one right identity. But, it may not be a real example for OP.

Answer 2

Consider for example the semigroup consisting of all constant functions on a set $X$ [acting on the right], together with one non-constant idempotent function $f$ (for example, let $f$ fix some point $x\in X$ and send every other point to some $y\neq x$). Then $f$ is a unique left identity, and $f$ is not a right identity.

In general I think it's probably helpful to think about this question in terms of transformation semigroups.

EDIT: Since this question has been sitting around with no accepted answer for a while, I'll state my last sentence a bit more strongly: You can determine exactly which transformation semigroups have a single left (or right) identity, and since every semigroup is isomorphic to a transformation semigroup, doing this will give you all examples. [Although I just noticed that the OP hasn't been on this site for about a month, so I guess the question might remain 'unanswered'.]