Monoid

A monoid is a set $S$ with an operation $\times$ which is associative and has an identity. That is, $M = (S, \times)$ is a monoid if and only if

$a \times (b \times c) = (a \times b)\times c$ for all $a, b, c \in S$

There is an element $e \in S$ such that $e\times a = a \times e = a$ for all $a \in S$ .

Alternatively, a monoid is a group without inverses.

Because the conditions on monoids are so weak, there are very few theorems of "monoid theory." However, monoids do arise from time to time in the study of abstract algebra, and many objects (such as all groups, as well as any ring with respect to either of its operations) are in fact monoids.

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Monoid