 Arity

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.

Arity is a concept that refers to the number of arguments or operands that a function or operation takes. It's commonly used in mathematics and programming to describe how many inputs a function requires to produce an output. For example: - A function with an arity of 0 takes no arguments (often referred to as a constant function). - A function with an arity of 1 takes one argument (e.g., a unary function).