# Are there any interesting semigroups that aren’t monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don’t have an identity element)?

To be a bit more precise, I guess I should ask if there are any interesting examples of semigroups $$(X, \ast)(X, \ast)$$ for which there is not a monoid $$(X, \ast, e)(X, \ast, e)$$ where $$ee$$ is in $$XX$$. I don’t consider an example like the set of real numbers greater than $$1010$$ (considered under addition) to be a sufficiently ‘natural’ semigroup for my purposes; if the domain can be extended in an obvious way to include an identity element then that’s not what I’m after.

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