ThmDex – An index of mathematical definitions, results, and conjectures.

▼	Set of symbols
▼	Alphabet
▼	Deduction system
▼	Theory
▼	Zermelo-Fraenkel set theory
▼	Set
▼	Binary cartesian set product
▼	Binary relation
▼	Map
▼	Operation
▼	N-operation
▼	Binary operation
▼	Enclosed binary operation
▼	Groupoid
▼	Semigroup

Definition D265

Monoid

Formulation 0

A D263: Groupoid $G = (X, +)$ is a monoid if and only if

(1)	\begin{equation} \forall \, x, y, z \in X : (x + y) + z = x + (y + z) \end{equation}
(2)	\begin{equation} \exists \, 0_G \in X : \forall \, x \in X : 0_G + x = x + 0_G = x \end{equation}

Formulation 1

A D263: Groupoid $G = (X, \times)$ is a monoid if and only if

(1)	\begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}
(2)	\begin{equation} \exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x \end{equation}

Formulation 2

A D263: Groupoid $G = (X, f)$ is a monoid if and only if

(1)	\begin{equation} \forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z)) \end{equation}
(2)	\begin{equation} \exists \, I \in X : \forall \, x \in X : f(I, x) = f(x, I) = x \end{equation}

Formulation 3

An D21: Algebraic structure $G = (X, \times)$ is a monoid if and only if

(1)	\begin{equation} \forall \, x, y \in X : x y \in X \end{equation}
(2)	\begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}
(3)	\begin{equation} \exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x \end{equation}

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