# Difference between revisions of "Algebraic operation"

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A mapping | A mapping | ||

− | $$\omega | + | $$\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 | + | 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"> | + | <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> | + | <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]] |

## 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