Finitary operation

Builds on Arity Function Cartesian product

Definition

Let $A$ be a set and let $n \ge 0$ be an integer. Write $A^n$ for the Cartesian product of $n$ copies of $A$ , with $A^0 = \set{()}$ the one-element set whose element is the empty tuple. An $n$ -ary operation on $A$ is a function $f \colon A^n \to A$ . A finitary operation on $A$ is an $n$ -ary operation on $A$ for some integer $n \ge 0$ , called its arity.

Remarks

Example: the map $f \colon \Z^3 \to \Z$ , $f(x, y, z) = x - y + z$ , is a finitary operation of arity $3$ on $\Z$ , because $x - y + z$ is an integer for all integers $x, y, z$ .

Non-example: division $(x, y) \mapsto x / y$ is not a binary operation on $\Q$ , because it is not defined when $y = 0$ , so it is not a function on all of $\Q^2$ . On $\Q \setminus \set{0}$ it is a binary operation.

The requirement that the values lie in $A$ is often called closure of $A$ under $f$ . Some authors use the word operation only for finitary operations.

Reading path

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Arity concept
Cartesian product concept
Function concept
Finitary operation definition

Tag	`K8RE4YQ1S9`
URI	`m2://K8RE4YQ1S9`
Kind	definition
Writing	W0 stub
MSC	08A62, 08A02
Defines	Finitary operation
Uses	Arity, Function, Cartesian product

Finitary operation

Definition

Remarks

Reading path

Further reading

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