Difference between revisions of "Algebraic operation"
From Encyclopedia of Mathematics
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A mapping | A mapping | ||
| − | + | $$\omega\colon A^n\to A$$ | |
| − | of the | + | 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|universal algebra]]. |
Revision as of 21:53, 16 March 2014
$n$-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] | 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 |
How to Cite This Entry:
Algebraic operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_operation&oldid=31376
Algebraic operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_operation&oldid=31376
This article was adapted from an original article by T.M. Baranovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article