View source for Algebraic operation

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''$n$-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 operation|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|universal algebra]].



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

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">   Józef Słomiński, "The theory of abstract algebras with infinitary operations"  ''Rozprawy Mat.'' , '''18'''  (1959). {{ZBL|0178.34104}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" (rev.ed.), Reidel  (1981)  pp. 13–14. {{ISBN|90-277-1213-1}}  {{ZBL|0461.08001}}</TD></TR>
</table>

[[Category:General algebraic systems]]