Definition:Monoid
Definition
A monoid is a semigroup with an identity element.
Monoid Axioms
The properties that define a monoid can be gathered together as follows:
A monoid is an algebraic structure $\struct {S, \circ}$ which satisfies the following properties:
\((\text S 0)\) | $:$ | Closure | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | |||||
\((\text S 1)\) | $:$ | Associativity | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | |||||
\((\text S 2)\) | $:$ | Identity | \(\ds \exists e_S \in S: \forall a \in S:\) | \(\ds e_S \circ a = a = a \circ e_S \) |
The element $e_S$ is called the identity element.
Also known as
Some treatments of group theory and abstract algebra do not introduce the term monoid, but simply discuss semigroups which happen to have an identity element.
Some sources present the monoid as $\struct {S, \circ, e_S}$ in order to place emphasis on the identity element, but this approach is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ as it complicates notation to little advantage.
Examples
Operation Defined as $x + y + x y$ on Real Numbers
Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:
- $\forall x, y \in \R: x \circ y := x + y + x y$
Then $\struct {\R, \circ}$ is a monoid whose identity is $0$.
Also see
- Definition:Group
- Monoid is not Empty, because it must at least have an identity element.
- Identity of Monoid is Unique: in a monoid there is only one identity element.
- Definition:Monoid Category, modelling monoids in category theory.
- Results about monoids can be found here.
Sources
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Definition $1.1 \text{(ii)}$
- 1981: Stanley Burris and H.P. Sankappanavar: A Course in Universal Algebra: $\text {II} \ \S 1$ Example $(2)$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): monoid: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): monoid
- 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): monoid
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.13$