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

A D263: Groupoid $G = (X, \times)$ is a semigroup if and only if $$\forall \, x, y, z \in X : (x y) z = x (y z)$$

An D21: Algebraic structure $G = (X, +)$ is a semigroup if and only if
 (1) $$\forall \, x, y \in X : x + y \in X$$ (2) $$\forall \, x, y, z \in X : (x + y) + z = x + (y + z)$$

An D21: Algebraic structure $G = (X, f)$ is a semigroup if and only if
 (1) $$\forall \, x, y \in X : f(x, y) \in X$$ (2) $$\forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z))$$

An D21: Algebraic structure $G = (X, \times)$ is a semigroup if and only if
 (1) $$\forall \, x, y \in X : x y \in X$$ (2) $$\forall \, x, y, z \in X : (x y) z = x (y z)$$
Children
 ▶ D265: Monoid