finitary

Finitary is an adjective used in mathematics, logic, and computer science to describe concepts that involve finite data, finite length, or finite arity. The term emphasizes that the object or construction is determined by finite components, in contrast to infinitary notions that allow infinite data or arity.

In universal algebra and related fields, a finitary operation has finite arity, meaning it takes a finite number of inputs. A finitary algebra or finitary theory uses only such finite-arity operations and finitely many variables in its defining equations. Infinitary variants, by contrast, permit operations of infinite arity or axioms with infinitely many variables. This distinction helps classify algebraic structures and their equational theories according to the complexity of their basic operations.

In category theory, a functor F is called finitary if it preserves filtered colimits. Equivalently, F is determined by its action on finite objects, and many finitary functors arise from algebraic theories. A common way to form finitary functors is by polynomial functors, which in Set have the form X ↦ ⊔_{i∈I} X^{n_i} with a finite index set I and finite arities n_i. Finitary functors are central to the study of finitary monads and their associated algebraic theories.

In logic and computer science, finitary also describes systems or rules that use only finite constructs, such as proofs with finitely many premises or derivations built from finite strings. The notion helps distinguish practical, constructible objects from those requiring infinite methods.

Etymology: from Latin finitus, meaning limited or finite. See also infinitary, arity, monad.