Monoid Definition

Question

A monoid is $(M,*,e)$. I am having trouble understanding what the 'multiplication' operation represents. Is it an operation for an element which returns another element or an operation which takes two elements and returns a third one?

Answer 1

The second: it represents an associative binary operation which takes elements, say, $a, b\in M$ and operates on them, and outputs a third (not necessarily distinct from a or b) value:

$$a * b = c, \;\quad a, b, c\in M$$

The symbol $*$ here is a "generic" symbol representing a binary operation which needs to be defined or specified: for example, addition of integers $(a + b)$, multiplication of positive integers $(a\cdot b \;\text{or just}\; ab)$.

Answer 2

The second is correct.

As an example you can consider the natural numbers $(\mathbb{N},+,0)$.