loop and quasigroup LoopAndQuasigroup 2002-09-06 18:53:46 2002-09-06 18:53:46 Definition loop quasigroup % 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 A \emph{quasigroup} is a groupoid $G$ with the property that for every $x, y \in G$, there are unique elements $w, z \in G$ such that $xw = y$ and $zx = y$. A \emph{loop} is a \emph{quasigroup} which has an identity element. What distinguishes a loop from a group is that the former need not satisfy the associative law.