multiplicatív

Multiplicativ is an adjective used in mathematics to describe objects, properties or operations that are related to multiplication. In general, a multiplicativ rule or structure preserves or interacts with the product of elements in a well-defined way.

In number theory, a function f: N → C is called multiplicativ if f(1)=1 and f(mn)=f(m)f(n) whenever gcd(m,n)=1. If f(mn)=f(m)f(n) for all positive integers m and n, the function is called completely multiplicativ. Classic examples of multiplicative functions include Euler’s totient φ(n), the divisor function d(n), and the Möbius function μ(n). Completely multiplicative examples include the Liouville function λ(n) and Dirichlet characters. These functions play a central role in multiplicative number theory, where values on prime powers and their independence under coprime factorization determine their values on all integers.

In algebra, multiplicativity often refers to maps that preserve multiplication. A ring homomorphism, for instance, preserves both addition and multiplication, sending the product of elements to the product of their images. Monoid homomorphisms preserve the multiplicative structure of monoids. The concept underpins many structural results, such as how multiplicative properties transfer through homomorphisms and how units and identity elements behave under mappings.

Beyond pure algebra, multiplicativity appears in probability (products of independent probabilities), in logic (multiplicative conjunctions in certain logics), and in category theory (monoidal structures that model tensor-like products). The term thus designates a broad family of notions tied to multiplication and its preservation.