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Difference between revisions of "Algebraic operation"

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A mapping
 
A mapping
  
$$\omega:A^n\to A$$
+
$$\omega\colon A^n\to A$$
  
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 $\omega:A^\alpha\to A$, 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]].
+
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]].
  
  
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====References====
 
====References====
 
<table>
 
<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>
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<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" , Reidel  (1981)  pp. 13–14</TD></TR>
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<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>
 
</table>
  
 
[[Category:General algebraic systems]]
 
[[Category:General algebraic systems]]

Latest revision as of 19:00, 26 April 2020

$n$-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
How to Cite This Entry:
Algebraic operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_operation&oldid=39753
This article was adapted from an original article by T.M. Baranovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article