Operations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 2 June 2012

Operations

An operation on a set $S$ is a function $$\begin{array}{cccc}\circ :& S\times S& ⟶& S\\ & (s_1,s_2)& ⟼& s_1\circ s_2\end{array}$$

An operation $\circ$ on a set $S$ is associative if $\circ :S\times S\to S$ satisfies $$\text{if}{s}_1,s_2,s_3\in S\text{then}(s_1\circ s_2)\circ s_3=s_1\circ (s_2\circ s_3).$$

An operation $\circ$ on a set $S$ is commutative if $\circ :S\times S\to S$ satisfies $$\text{if}{s}_1,s_2\in S\text{then}{s}_1\circ s_2=s_2\circ s_1.$$

Examples

The function $$\begin{array}{rcll}+:& \mathbb{Z}\times \mathbb{Z}& ⟶& \mathbb{Z}\\ & (i,j)& ⟼& i+j\end{array}$$ is an operation. This operation is both commutative and associative.

The function $$\begin{array}{rcll}-:& \mathbb{Z}\times \mathbb{Z}& ⟶& \mathbb{Z}\\ & (i,j)& ⟼& i-j\end{array}$$ is an operation. This operation is both noncommutative and nonassociative.

Notes and References

Since an operation is just a function, perhaps the term operation should be deprecated.

References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

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