\documentclass[a4paper, 10pt]{article}
\usepackage[estonian]{babel}
\usepackage{t1enc}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{rotating}
\usepackage{hyperref}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\renewcommand{\P}{\mathbb{P}}

\topmargin-8em
\oddsidemargin0cm
\textwidth16cm
\textheight32cm
\evensidemargin-2cm

\begin{document}

\pagestyle{empty}

\large

{\hfill{Fall 2021} \smallskip %
\vspace{2em}

\begin{center}
{\LARGE Introduction to Algebraic Structures\\} 

{\LARGE Problem Set 5\\}
\vspace{1em}
{\LARGE Semigroups.}
\end{center}
}


\Large
\bigskip\smallskip

\bigskip

\noindent 1. Prove that in a finite monoid all elements have right inverses if and only if they have left inverses. 

\bigskip

\noindent 2. Let $(S,\cdot)$ be a semigroup satisfying the identity $st^2s=s^2$. Define a new binary operation $*:S\times S\to S$ by $s*t = t^2s.$ Is $(S,*)$ a semigroup? Is it a group? Why?

\bigskip

\noindent 3. Find an infinite semigroup which is not a group but contains a proper subgroup. 

\smallskip

\noindent 4. Prove that a semigroup is left simple and right simple if and only if it is a group.

\smallskip

\noindent 5. Let $S$ be a semigroup with the following Cayley table:
\vskip-10pt
$$\begin{array}{|c|ccccccc|}
\hline
S & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
\hline
1 & 1 & 1 & 1 & 1 & 1 & 7 & 7\\
2 & 1 & 1 & 1 & 1 & 1 & 7 & 7\\
3 & 1 & 1 & 1 & 2 & 1 & 7 & 7\\
4 & 1 & 1 & 2 & 1 & 1 & 7 & 7\\
5 & 5 & 5 & 5 & 5 & 5 & 6 & 6\\
6 & 5 & 5 & 5 & 5 & 5 & 6 & 6\\
7 & 1 & 1 & 1 & 1 & 1 & 7 & 7\\
\hline
\end{array}$$
\vskip-2pt
\noindent Write out all the $\mathcal{D}$-classes of $S$ and their distribution into $\mathcal{L}$- and $\mathcal{R}$-classes (the ``egg-box'' diagram). 

\bigskip

\noindent 6. Prove that the $\mathcal{H}$-relation on a Rees matrix semigroup $\mathcal{M} = \mathcal{M}[G,I,\Lambda,P]$ over a group $G$ can be described thus:
$$(i,g,\lambda)\;\mathcal{H}\;(j,h,\mu)\Longleftrightarrow i=j\wedge \lambda = \mu.$$

\noindent 7. Prove that $\mathcal{D} = \mathcal{J}$ in any finite semigroup. 

\smallskip

\noindent 8. Prove that if a semigroup $S$ satisfies the identity $s^at^b=ts$ for some $a,b\geq 2$, then $S$ is commutative. 

\bigskip

\noindent 9${^*}$. Prove or disprove: a cancellative semigroup $S$ can always be embedded in a group, i.e. there exist a group $G$ and an injective homomorphism $\iota:S\to G$. 

\bigskip

\noindent 10${^{*}}$. Let $S$ be a regular semigroup the idempotents of which form a rectangular band. Prove that $S$ is isomorphic to the direct product of a group and a rectangular band. 

\end{document}