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 D558
Ringoid
Formulation 0
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)
Formulation 1
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)
Formulation 2
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 $+$
Children
▶ D559: Semiring