Mathematical Structures: Regular semigroups

# Regular semigroups

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http://mathcs.chapman.edu/structuresold/files/Regular_semigroups.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Regular semigroups}

\abbreviation{RSgrp}
\begin{definition}
An element $x$ of a semigroup $S$ is said to be \emph{regular} if exists $y$ in $S$ such that $xyx=x$.
\end{definition}

\begin{definition}
A \emph{regular semigroup} is a \href{Semigroups.pdf}{semigroups} $\mathbf{S}=\left\langle S,\cdot \right\rangle$ such that
each element is regular.
\end{definition}

\begin{definition}
A \emph{regular semigroup} is a structure $\mathbf{S}=\left\langle S,\cdot \right\rangle$, where $\cdot$ is an infix binary operation, called
the \emph{semigroup product}, such that

$\cdot$ is associative:  $(xy)z=x(yz)$

each element is \emph{regular}:  $\exists y(xyx=x)$
\end{definition}

\begin{definition}
We say that $y$ is an \emph{inverse} of an element $x$ in a semigroup $S$ if $x=xyx$ and $y=yxy$.
\end{definition}

\begin{morphisms}
Let $\mathbf{S}$ and $\mathbf{T}$ be regular semigroups. A morphism from
$\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
homomorphism:

$h(xy)=h(x)h(y)$

\end{morphisms}
\begin{basic_results}
If $x$ is a regular element of a semigroup (say $x=xyx$), then $x$ has an inverse, namely $yxy$, since $x=x(yxy)x$ and $yxy=(yxy)x(yxy)$.
\end{basic_results}
\begin{examples}
\begin{example}
$\langle T_X,\circ\rangle$, the \emph{full transformation semigroup} of functions on $X$, with composition.

$\langle End(V),\circ\rangle$, the \emph{endomorphism monoid} of a vector space $V$, with composition.

\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & First-order\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & No\\\hline
Residual size & \\\hline
Congruence distributive & No\\\hline
Congruence modular & \\\hline
Congruence n-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & No\\\hline
Amalgamation property & No\\\hline
Strong amalgamation property & No\\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &3\\ f(3)= &9\\ f(4)= &42\\ f(5)= &206\\ f(6)= &1352\\ f(7)= &10168\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{Bands.pdf}{Bands}

\href{Inverse_semigroups.pdf}{Inverse semigroups}

\href{Completely_regular_semigroups.pdf}{Completely regular semigroups}

\end{subclasses}
\begin{superclasses}\

\href{Semigroups.pdf}{Semigroups}

\end{superclasses}

\begin{bibdiv}
\begin{biblist}

\bib{MR1455373}{book}{
author={Howie, John M.},
title={Fundamentals of semigroup theory},
series={London Mathematical Society Monographs. New Series},
volume={12},
note={Oxford Science Publications},
publisher={The Clarendon Press Oxford University Press},
place={New York},
date={1995},
pages={x+351},
isbn={0-19-851194-9},
review={\MR{1455373 (98e:20059)}},
}

\end{biblist}
\end{bibdiv}

\end{document}
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