Are there any interesting semigroups that aren't monoids?

Convolution of functions/distributions is useful in a variety of fields, and the identity element, the dirac delta, is not strictly a function.

One source of monoids is given by taking rings with identity, and forgetting about addition. So similarly, one source of semigroups that are not monoids is taking rings without identity, and forgetting about addition. With this in mind, let me explain one basic source of rings without identity.

A basic source of rings is given by taking functions satisfying some reasonable condition on a space, e.g. continuous real or complex valued functions on a space, with pointwise addition and multiplication. Of course, the constant function 1 is continuous, and so this gives a ring with identity.

But suppose now that we impose some condition, such as "all functions that are continuous, and which furthermore vanish at some specified point". This throws out the constant function 1, and so gives a ring without identity. Now you could naturally object that this is artificial (as per the requirement in the question that there not be an obvious extension to a monoid), so let me add more explanation as to why it need not be.

One example of a point to consider is "the point at infinity", i.e. we could look at all functions which vanish at infinity, i.e. which on the complement of larger and larger compact sets, grow smaller and smaller. This is a natural condition to impose in many analytic contexts, and so gives a natural example. (The reason that this kind of growth condition is natural in analysis is that, on a non-compact space, e.g. the real line, a random continuous function may not be integrable (just as an example), and imposing some decay at infinity (perhaps of the kind I specified, or perhaps something more quantitive) becomes a way to rescue the situation.) (Note also that the example that Tomer Vromen gives is exactly of this form.)

Finally, note that if your semigroup doesn't have an identity, then you can always formally adjoin one, just by throwing in an extra element $e$ and declaring that $ex = x$ for all $x$.

One can do a similar thing for rings without identity. If $A$ is a $C$-algebra (say) without an identity, then one can form $A \oplus C e$ (the direct sum), and declare that $e$ acts as a multiplicative identity. This is a frequently-used technique in the theory of rings-without-identity.

P.S. I don't know much literature about semigroups without identity, but for rings without identity, the best literature I know of is in functional analysis books; e.g. Naimark's classic Normed Rings often treats the case of Banach algebras (and the like) without identity in addition to the case when they do have identity, exactly so as to be able to handle examples such as the ring of continuous functions on a locally compact space that vanish at infinity.

Here is a cheap answer which takes the stance that semigroups are inherently interesting :) Consider the following definition of a group:

$\textbf{Definition}$ A semigroup $S$ is said to be a group if the following hold:

1. There is an $e \in S$ such that $ea=a$ for all $a\in S$
2. For each $a \in S$ there is an element $a^{-1} \in S$ with $a^{-1}a=e$

At one point in my life, it seemed natural to ask what happens if we replace axiom 2 with the very similar axiom

2$^\prime$. For each $a \in S$ there is an element $a^{-1} \in S$ with $aa^{-1} = e$.

It is a fun exercise to work out some of the consequences that result from this. Here are a few facts about a semigroup $S$ which satisfies $1$ and $2^\prime$:

• If $e$ is the unique element of $S$ satisfying axiom 1, then $S$ is a group
• If $S$ has an identity (in the usual sense) then $S$ is a group
• The principal left ideal $Sa = \{sa \mid s \in S\}$ is a group for all $a \in S$, and in fact all principal left ideals of $S$ are isomorphic as groups.

It is not difficult to find examples of such semigroups that are not groups. For example, consider the following set of $2\times 2$ matrices (with matrix multiplication as the operation): $$\left\{\begin{pmatrix} a & b \\ 0 & 0\end{pmatrix} \mid a,b \in \mathbb{R}, a \neq 0\right\}$$ Or, an example that appears as exercise 30 in section 4 of Fraleigh's abstract algebra text: the nonzero real numbers under the operation $\ast$ defined by $a\ast b = |a|b$.

Certainly it is debatable whether or not semigroups satisfying axioms 1 and 2$^\prime$ are "interesting" or "natural". But I guess I think they are. And, I am not the only one (or the first one, by a long shot!) to think this. See Mann, On certain systems which are almost groups (MR). (A bit of googling will turn up more results if you are interested.)