ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D265
Monoid

A D263: Groupoid $G = (X, +)$ is a monoid if and only if
 (1) $$\forall \, x, y, z \in X : (x + y) + z = x + (y + z)$$ (2) $$\exists \, 0_G \in X : \forall \, x \in X : 0_G + x = x + 0_G = x$$

A D263: Groupoid $G = (X, \times)$ is a monoid if and only if
 (1) $$\forall \, x, y, z \in X : (x y) z = x (y z)$$ (2) $$\exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x$$

A D263: Groupoid $G = (X, f)$ is a monoid if and only if
 (1) $$\forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z))$$ (2) $$\exists \, I \in X : \forall \, x \in X : f(I, x) = f(x, I) = x$$

An D21: Algebraic structure $G = (X, \times)$ is a monoid if and only if
 (1) $$\forall \, x, y \in X : x y \in X$$ (2) $$\forall \, x, y, z \in X : (x y) z = x (y z)$$ (3) $$\exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x$$
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