Definition:Monoid
From ProofWiki
Definition
A semigroup with an identity element is called a monoid.
That is, a monoid is an algebraic structure $\left({S, \circ, e_S}\right)$ which satisfies the following three properties:
| Closure: | $\forall a, b \in S: a \circ b \in S$. |
| Associativity: | $\forall a, b, c \in S: \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$. |
| Identity: | $\exists e_S \in S: \forall a \in S: a \circ e_S = a = e_S \circ a$. |
The element $e_S$ is called the identity.
From Identity of Semigroup is Unique, there can only be one such identity element.
A monoid can not be empty, because it must at least have an identity.
Also see
- Results about monoids can be found here.
Sources
- Stanley Burris and H. P. Sankappanavar: A Course in Universal Algebra (1981): $\text {II} \ \S 1$ Example $(2)$
