Monoids

A monoid is made up of a set or domain $T$ and a binary operator $z = f(x, y)$ operating on the same domain, $T \times T \rightarrow T$. This binary operator must be associative, that is $f(a, f(b, c)) = f(f(a, b), c)$ is always true. Associativity is important for operations like reduce and the multiplication step of mul!.

The operator is also be equipped with an identity such that $f(x, 0) = f(0, x) = x$. Some monoids are equipped with a terminal or annihilator such that $z = f(z, x) \forall x$.

Monoids are used primarily in the reduce operation. Their other use is as a component of semirings in the mul! operation.

Built-Ins

Monoid{F, I, T}