identity element IdentityElement 2002-06-27 19:18:27 2002-10-10 18:33:59 Definition group % 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{multiplicative} Let $G$ be a groupoid, that is a set with a binary operation $G \times G \to G$, written muliplicatively so that $(x, y) \mapsto xy$. An \emph{identity element} for $G$ is an element $e$ such that $ge = eg = g$ for all $g \in G$. The symbol $e$ is most commonly used for identity elements. Another common symbol for an identity element is $1$, particularly in semigroup theory (and ring theory, considering the multiplicative structure as a semigroup). Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element.