Monoidlike

Monoidlike is an informal mathematical adjective used to describe objects, structures, or constructions that resemble a monoid in essential respects, namely the presence of a binary operation that is associative and a distinguished unit, within a given ambient setting. In the standard set-theoretic sense, a monoid is a set M with a binary operation •: M×M→M that is associative and has an identity element e with e•m = m = m•e. Monoidlike objects generalize this idea to other contexts where the underlying "set" is replaced by an object in a category, or where the operation is internal to a structure. For example, a monoid object in a monoidal category (C, ⊗, I) consists of an object M with morphisms μ: M⊗M→M and η: I→M satisfying associativity and unitality strictly; such structures are often referred to as monoid-like in general discussions.

Other uses: The endomorphism object End(A) in a suitable category can carry a monoid structure under composition, making it monoidlike. In computer science, monoidlike data types are those that support a binary operation with an identity that occurs in functional programming and type theory.

Notes: The precise meaning of monoidlike is context-dependent. It typically denotes a relaxation or generalization of the monoid axioms, used to indicate a structure that behaves as a monoid with respect to the relevant notions of product and unit in that setting.

See also: Monoid, Monoid object, Monoidal category, Endomorphism monoid.