**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) |