subsemigroup,, submonoid,, and subgroup SubmonoidSubsemigroup 2002-09-06 17:57:36 2002-09-08 00:27:17 Definition subsemigroup submonoid subgroup % 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 Let $S$ be a semigroup, and let $T$ be a subset of $S$. $T$ is a \emph{subsemigroup} of $S$ if $T$ is closed under the operation of $S$; that it if $xy \in T$ for all $x, y \in T$. $T$ is a \emph{submonoid} of $S$ if $T$ is a subsemigroup, and $T$ has an identity element. $T$ is a \emph{subgroup} of $S$ if $T$ is a submonoid which is a group. Note that submonoids and subgroups do not have to have the same identity element as $S$ itself (indeed, $S$ may not have an identity element). The identity element may be any idempotent element of $S$. Let $e \in S$ be an idempotent element. Then there is a maximal subsemigroup of $S$ for which $e$ is the identity: $$eSe = \{ exe \mid x \in S \}.$$ In addition, there is a maximal subgroup for which $e$ is the identity: $$\mathcal{U}(eSe) = \{x \in eSe \mid \exists y \in eSe \;\text{st}\; xy=yx=e \}.$$ Subgroups with different identity elements are disjoint. To see this, suppose that $G$ and $H$ are subgroups of a semigroup $S$ with identity elements $e$ and $f$ respectively, and suppose $x \in G \cap H$. Then $x$ has an inverse $y \in G$, and an inverse $z \in H$. We have: $$e = xy = fxy = fe = zxe = zx = f.$$ Thus intersecting subgroups have the same identity element.