multiplicativus

Multiplicativus is a term encountered in abstract algebra and related areas to denote a map or function that preserves multiplication across its domain. The name is Latin in origin, meaning related to multiplication, and in some contexts it is used interchangeably with the notion of a multiplicative or completely multiplicative function. The exact usage can vary by author, but the core idea is a structure-preserving rule for products.

Definition and scope: Let M and N be algebraic structures equipped with a binary operation of multiplication, denoted by ·. A function f: M → N is called multiplicativus if f(x · y) = f(x) · f(y) for all x, y ∈ M. If this condition holds for all elements in M and N, f is often described as completely multiplicativus. When the condition is required only in certain restricted contexts (for example, in specific substructures or under additional constraints), the term multiplicativus is sometimes used to indicate that weaker multiplicativity holds.

Properties: A multiplicativus map preserves the multiplicative structure of the domain in the codomain. If f and g are multiplicativus and their domains and codomains are compatible, their composition can also be multiplicativus. The concept aligns with that of monoid homomorphisms when the underlying structures have a multiplicative monoid operation, and it interacts with other structure-preserving maps, such as additive homomorphisms, in the broader study of algebraic structures.

Examples: The identity map on any multiplicative structure is completely multiplicativus. For a fixed integer k, the map f(n) = n^k on the integers is completely multiplicativus since f(xy) = (xy)^k = x^k y^k. The zero map sending every element to the absorbing element (e.g., 0 in a multiplicative monoid) is also multiplicativus, as f(xy) = 0 = 0 · 0 = f(x) · f(y).

Usage and context: In many texts, the standard terms are multiplicative or completely multiplicative functions, while multiplicativus appears as an alternative wording or in more general or fictional discussions. The precise meaning depends on the adopted conventions of the author or area of study.

See also: multiplicative function, completely multiplicative function, monoid homomorphism.