Multiplicationsis

Multiplicationsis is a hypothetical framework in mathematics that treats multiplication as the primary structural operation to be studied across a range of algebraic systems. In this view, the multiplicative aspect of a structure—its way of combining elements in a binary operation—is foregrounded, while addition and other operations are considered secondary or derived in some contexts.

Formal foundations: The core objects are sets equipped with a binary operation often denoted by ·, subjected to axioms that may vary by variant: associative or non-associative, with or without identity, with partial or total closure. The theory investigates how multiplicative properties interact with other structure, including distributivity, invertibility, and compatibility with maps between systems (multiplicative homomorphisms).

Origins and usage: The term emerged in philosophical discussions about arithmetic foundations; it has no single canonical formulation and remains unofficial. It is used mainly in thought experiments, pedagogical contexts, and speculative expositions to contrast multiplicative structure across algebraic categories.

Key notions: Multiplicative semigroups, monoids, and groups provide canonical examples; multiplicative inverses; units; zero divisors; multiplicative order; and generalized distributive laws form core topics. Researchers may investigate invariants preserved under multiplicative maps or classify systems by their multiplicative profile.

Relation to established algebra: Multiplicationsis is not a separate discipline but a lens for comparing algebraic structures. Critics note redundancy with existing concepts like semigroups and rings; proponents argue it emphasizes multiplicative phenomena that are not easily captured by additive-first viewpoints.

See also: multiplication, distributive law, semigroup, monoid, ring, algebra, category theory, multiplicative homomorphism.