Algebraic operation
-ary operation, on a set
A mapping
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of the -th Cartesian power of the set
into the set
itself. The number
is known as the arity of the algebraic operation. Historically, the concepts of binary
and unary (
) operations were the first to be considered. Nullary
operations are fixed elements of the set
; they are also known as distinguished elements or constants. In the 20th century the concept of an infinitary operation appeared, i.e. a mapping
, where
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] | J. Stominski, "The theory of abstract algebras with infinitary operations" Rozprawy Mat. , 18 (1959) |
[a2] | P.M. Cohn, "Universal algebra" , Reidel (1981) pp. 13–14 |
Algebraic operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_operation&oldid=12966