Algebraic operation

$n$-ary operation, on a set $A$

A mapping

$$\omega\colon A^n\to A$$

of the $n$-th Cartesian power of the set $A$ into the set $A$ itself. The number $n$ is known as the arity of the algebraic operation. Historically, the concepts of binary $(n=2)$ and unary ($n=1$) operations were the first to be considered. Nullary $(n=0)$ operations are fixed elements of the set $A$; they are also known as distinguished elements or constants. In the 20th century the concept of an infinitary operation appeared, i.e. a mapping $\omega\colon A^\alpha\to A$, where $\alpha$ is an arbitrary cardinal number. A set with a system of algebraic operations defined on it is called a universal algebra.

Comments

The study of infinitary operations actually started in the late 1950s [a1]. A nullary operation is also called a noughtary operation [a2].

References

[a1]	Józef Słomiński, "The theory of abstract algebras with infinitary operations" Rozprawy Mat. , 18 (1959). Zbl 0178.34104
[a2]	P.M. Cohn, "Universal algebra" (rev.ed.), Reidel (1981) pp. 13–14. ISBN 90-277-1213-1 Zbl 0461.08001