Definition. Monoid [dm-000O]

Liam O'Connor

A monoid $(S, \iota , \bullet )$ consists of a set $S$ , an identity element $\iota \in S$ , and a binary operator $(\bullet ) : S \times S \rightarrow S$ which satisfies, for all $x,y,z \in S$ :

associativity: $x \bullet (y \bullet z) = (x \bullet y) \bullet z$
left identity: $\iota \bullet x = x$
right identity: $x \bullet \iota = x$