| (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) | (R3) $\forall \, x, y, z \in X : g(x, f(y, z)) = f(g(x, y), g(x, z))$ (D555: Left-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) |
| (7) | $\forall \, x, y \in X : f(x, y) = f(y, x)$ (D489: Commutative binary operation) |
| (8) | $\exists \, 0_R \in X : \forall \, x \in X : f(0_R, x) = f(x, 0_R) = x$ (D39: Identity element) |
| (9) | $\forall \, x \in X : \exists \, {-x} \in X: f(-x, x) = f(x, -x) = 0_R$ (D40: Inverse element) |