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Difference between revisions of "Algebraic operation"

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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stominski,  "The theory of abstract algebras with infinitary operations"  ''Rozprawy Mat.'' , '''18'''  (1959)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)  pp. 13–14</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stominski,  "The theory of abstract algebras with infinitary operations"  ''Rozprawy Mat.'' , '''18'''  (1959)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)  pp. 13–14</TD></TR>
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[[Category:General algebraic systems]]

Revision as of 22:17, 26 October 2014

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

A mapping

$$\omega:A^n\to A$$

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 $\omega:A^\alpha\to A$, 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=31378
This article was adapted from an original article by T.M. Baranovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article