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
Definition D264
Semigroup
Formulation 0
A D263: Groupoid $G = (X, \times)$ is a semigroup if and only if \begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}
Formulation 1
An D21: Algebraic structure $G = (X, +)$ is a semigroup 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}
Formulation 2
An D21: Algebraic structure $G = (X, f)$ is a semigroup if and only if
(1) \begin{equation} \forall \, x, y \in X : f(x, y) \in X \end{equation}
(2) \begin{equation} \forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z)) \end{equation}
Formulation 3
An D21: Algebraic structure $G = (X, \times)$ is a semigroup 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}
Children
D265: Monoid