associativelike

Associativelike is an informal descriptor used to indicate a binary operation that resembles the associative law but does not satisfy it in full generality. It is not a standard algebraic axiom, but a heuristic term encountered in discussions of weakened or context-dependent forms of associativity, such as partial associativity, quasi-associativity, or associativity up to isomorphism.

Forms typically encompassed by the idea of associativelike include:

- Partial associativity: for a subset T of the underlying set, the law (a*b)*c = a*(b*c) holds for all a, b, c in T, while outside of T the equality may fail.

- Quasi-associativity: the associativity law holds up to a fixed transformation or perturbation, so (a*b)*c and a*(b*c) are related by a predictable rule.

- Associativity up to isomorphism: in categorical contexts (for example, monoidal categories), the tensor product is strictly associative only up to a natural isomorphism, which is a controlled relaxation of strict associativity.

Examples are often synthetic or contextual. A binary operation may be designed to be associative within a chosen substructure but not across the entire set, or it may be strictly associative after applying a certain equivalence relation. In category theory, associativity is frequently understood in terms of structure-preserving maps (associators) that satisfy coherence conditions, which is a formal way of expressing associativity in a weaker sense.

Relation to other concepts: associativelike is related to quasi-groups and loops (which may lack associativity), as well as to the broader study of algebraic structures with weakened laws. Because its meaning is not standardized, precise definitions are provided whenever the term is used.

See also: associativity, partial associativity, quasi-associativity, monoidal category, associator.