multiplicationare

Multiplicationare is a term used in some mathematical texts and thought experiments to denote a family of binary operations that generalize or modify ordinary multiplication. It is not a standard operation in mainstream mathematics, but a flexible concept used in didactic and speculative discussions. A multiplicationare operation, denoted here by ⊗, is a binary operation on a set S that is studied alongside the usual addition and multiplication on S.

Formally, ⊗: S×S → S is defined on a chosen set with properties that may vary by variant. Common variants require closure; associativity; and sometimes commutativity. Some formulations also include a multiplicative identity e with a ⊗ e = e ⊗ a = a for all a in S, while others are nonunital. Distributivity over addition can be imposed or omitted, yielding bilinear-like structures or more general magma structures.

A simple concrete variant on the integers defines a ⊗ b = ab + a + b. This operation is commutative and associative but has no multiplicative identity, illustrating how altering the rule changes the algebraic structure while preserving some familiar behaviors.

Variants of multiplicationare are used to explore how different axioms affect structure, to illustrate nonstandard arithmetics in pedagogical contexts, and in fictional or hypothetical mathematical systems. In algebraic terms, such operations can lead to ring-like or semiring-like structures when paired with addition, but they need not satisfy all ring axioms.

See also: binary operation, semigroup, monoid, ring.