commutative semigroup CommutativeSemigroup 2002-11-05 18:49:09 2002-11-05 18:49:09 Definition commutative commutative monoid % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{term} \PMlinkescapeword{identity} A semigroup $S$ is \emph{commutative} if the defining binary operation is \PMlinkname{commutative}{Commutative}. That is, for all $x, y \in S$, the identity $xy = yx$ holds. Although the term \emph{Abelian semigroup} is sometimes used, it is more common simply to refer to such semigroups as \emph{commutative semigroups}. A monoid which is also a commutative semigroup is called a \emph{commutative monoid}.