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 $(X, \ast)$ for which there is not a monoid $(X, \ast, e)$ where $e$ is in $X$. I don't consider an example like the set of real numbers greater than $10$ (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.
Best Answer
Convolution of functions/distributions is useful in a variety of fields, and the identity element, the dirac delta, is not strictly a function.