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 ▼ Ringoid ▼ Semiring ▼ Ring ▼ Multiplicative group ▼ Multiplicative monoid ▼ Multiplicative semigroup ▼ Multiplicative groupoid ▼ Multiplicative binary operation ▼ Natural number multiplication operation ▼ Integer multiplication operation ▼ Rational multiplication operation
Definition D610
Real multiplication operation

Let $\mathbb{R}$ be the D282: Set of real numbers.
Let $\cdot$ be the D609: Rational multiplication operation.
A D554: Binary operation $* : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a real multiplication operation if and only if
 (1) $$\forall \, x, y \in [0, \infty) : x * y = 0 \cup \{ a b : 0 \leq a \in x \text{ and } 0 \leq b \in y \}$$ (2) $$\forall \, x, y \in (- \infty, 0) : x * y = (x \cup - x) * (y \cup - y)$$ (3) $$\forall \, x \in [0, \infty) : \forall \, y \in (- \infty, 0) : x * y = - (x \cup - x) * (y \cup - y)$$
Children
 ▶ Complex multiplication operation