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In universal algebra, a finitary algebraic operation on a set $S$ is a functionfor a nonnegative integer $n$, its arity. A binary operation has $n=2$, as with addition on the integers. A unary operation has $n=1$, as with taking the additive inverse $a\mapsto -a$.

A nullary operation has $n=0$: since $S^0$ contains the single empty tuple, such an operation selects a distinguished element of $S$, such as zero in a ring. Thus constants are operations too. The fixed codomain $S$ expresses closure, while additional identities such as the associative property specify laws an operation may satisfy.

The arity of a finitary algebraic operation is the number $n$ of its input positions. An operation of arity $n$ on $S$ is a function $S^n\to S$. A nullary operation chooses a constant, a unary operation takes one input, and a binary operation takes two.

For example, addition on the integers is binary even though repeated addition allows an expression such as $a+b+c$. That expression composes two binary operations; associativity is what makes its parenthesization irrelevant. A symbol taking arbitrarily many inputs would need a different specification, such as one operation for each $n$.

A fixed arity is part of an algebraic signature and determines how terms are built and which inputs a homomorphism must preserve. Operations of the same arity can nevertheless satisfy completely different identities.

Universal algebra studies algebraic structures by specifying their operations and identities in a common language. A signature assigns an arity to each operation symbol. A structure interprets each symbol as a function $A^n\to A$ on its underlying set; an identity requires two terms to agree for every assignment of their variables.

For example, a group can be presented with a binary operation, a unary operation for inversion, and a nullary operation for its identity element. A ring adds multiplication and its distributive identities. These signatures differ, but the same definitions of a structure-preserving homomorphism, a compatible equivalence relation, and a quotient structure apply to both.

This shared framework explains why many proofs recur across algebra: a homomorphism identifies elements in a way compatible with every specified operation. The resulting quotient must preserve all those operations, not only a selected multiplication table. Universal algebra isolates that compatibility before considering special properties of groups, rings or other examples.