Ringoid

An D21: Algebraic structure $R = (X, f, g)$ is a ringoid if and only if
 (1) $\forall \, x, y \in X : f(x, y) \in X$ (D20: Enclosed binary operation) (2) (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)

An D21: Algebraic structure $R = (X, +, \times)$ is a ringoid 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)

An D21: Algebraic structure $R = (X, +, \times)$ is a ringoid if and only if
 (1) $G = (X, +)$ is a D263: Groupoid (2) $H = (X, +)$ is a D263: Groupoid (3) The D20: Enclosed binary operation $\times$ is a D557: Distributive binary operation over $+$
Child definitions