ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D559
Semiring

An D21: Algebraic structure $R = (X, +, \times)$ is a semiring if and only if
(1) $\forall \, x, y \in X : x + y \in X$ (D20: Enclosed binary operation)
(2) $\forall \, x, y \in X : x y \in X$ (D20: Enclosed binary operation)
(3) $\forall \, x, y, z \in X : x (y + z) = (x y) + (x z)$ (D555: Left-distributive binary operation)
(4) $\forall \, x, y, z \in X : (x + y) z = (x z) + (y z)$ (D556: Right-distributive binary operation)
(5) $\forall \, x, y, z \in X : x + (y + z) = (x + y) + z$ (D488: Associative binary operation)
(6) $\forall \, x, y, z \in X : x (y z) = (x y) z$ (D488: Associative binary operation)

An D21: Algebraic structure $R = (X, +, \times)$ is a semiring if and only if
 (1) $G = (X, +)$ is a D264: Semigroup (2) $H = (X, \times)$ is a D264: Semigroup (3) The D20: Enclosed binary operation $\times$ is a D557: Distributive binary operation over $+$

An D21: Algebraic structure $R = (X, f, g)$ is a semiring if and only if
(1) $\forall \, x, y \in X : f(x, y) \in X$ (D20: Enclosed binary operation)
(2) $\forall \, x, y \in X : g(x, y) \in X$ (D20: Enclosed binary operation)
(3) $\forall \, x, y, z \in X : g(x, f(y, z)) = f(g(x, y), g(x, z))$ (D555: Left-distributive binary operation)
(4) $\forall \, x, y, z \in X : g(f(x, y), z) = f(g(x, z), g(y, z))$ (D556: Right-distributive binary operation)
(5) $\forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z))$ (D488: Associative binary operation)
(6) $\forall \, x, y, z \in X : g(g(x, y), z) = g(x, g(y, z))$ (D488: Associative binary operation)
Children
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