Operations

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

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

Last updates: 22 September 2009

Definitions

An operation on a set $S$ is a map $\circ :S\times S\to S$. If $s_1,s_2\in S$ we write $s_1\circ s_2$ instead of $\circ \left(\left(s_1,s_2\right)\right)$.

An operation on a set $S$ is associative if, for all $s_1,s_2,s_3\in S$, $$\left(s_1\circ s_2\right)\circ s_3=s_1\circ \left(s_2\circ s_3\right)\text{.}$$

An operation on a set is commutative if, for all $\left(s_1,s_2\right)\in S$, $$s_1\circ s_2=s_2\circ s_1\text{.}$$

Examples

The map $+:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ given by $$\begin{array}{rcl}+:\mathbb{Z}\times \mathbb{Z}& ⟶& \mathbb{Z}\\ \left(i,j\right)& ⟼& i+j\end{array}$$ is an operation. This operation is both commutative and associative.

The map $-:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ given by $$\begin{array}{rcl}-:\mathbb{Z}\times \mathbb{Z}& ⟶& \mathbb{Z}\\ \left(i,j\right)& ⟼& i-j\end{array}$$ is an operation. This operation is both noncommutative and nonassociative.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)