commutative Commutative 2002-02-18 21:51:51 2008-12-21 13:16:49 Definition non-commutative \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{psfrag} %\usepackage{graphicx} %\usepackage{xypic} Let $S$ be a set and $\circ$ a binary operation on it. $\circ$ is said to be \emph{commutative} if $$a \circ b = b \circ a$$ for all $a,b \in S$. Viewing $\circ$ as a function from $S\times S$ to $S$, the commutativity of $\circ$ can be notated as $$\circ(a,b)=\circ(b,a).$$ Some common examples of commutative operations are \begin{itemize} \item addition over the integers: $m+n=m+n$ for all integers $m,n$ \item multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$ \item addition over $n \times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$, and \item multiplication over the reals: $rs=sr$, for all real numbers $r,s$. \end{itemize} A binary operation that is not commutative is said to be \emph{non-commutative}. A common example of a non-commutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general, $$a-b\ne b-a.$$ For instance, $2-1=1\ne -1 = 1-2$. Other examples of non-commutative binary operations can be found in the attachment below. \textbf{Remark}. The notion of commutativity can be generalized to $n$-ary operations, where $n\ge 2$. An $n$-ary operation $f$ on a set $A$ is said to be \emph{commutative} if $$f(a_1,a_2,\ldots, a_n)=f(a_{\pi(1)},a_{\pi(2)},\ldots, a_{\pi(n)})$$ for every permutation $\pi$ on $\lbrace 1,2,\ldots, n\rbrace$, and for every choice of $n$ elements $a_i$ of $A$.