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.

**Answer**

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

**Attribution***Source : Link , Question Author : Community , Answer Author : Calvin Khor*