semigroups.tex

\subsection*{Extension: Less Than A Group}


It turns out that there are algebraic objects which are not quite 
groups, but which are still of interest.

\begin{definition}
  A \defn{semigroup}{semigroup} is a set with an associative binary 
  operation.
  
  A \defn{monoid}{monoid} is a semigroup with an identity element $e$.
\end{definition}

As one can see, the bar for being a semigroup is extremely low, and 
one is not asking that much more to get a monoid.

\begin{proposition}\label{prop:semigroup2group}
  A semigroup $(G, \ast)$ is a group if and only if there is a left 
  identity $e \in G$, and for every element $x \in G$ there is an 
  element $x^{-1} \in G$ which is a left-inverse (ie.\ $x^{-1}x = e$).
\end{proposition}

\begin{proposition}\label{prop:semigroup2group2}
    A semigroup $(G,\ast)$ is a group if and only if for all $a$, $b 
    \in G$, one can find elements $x$ and $y \in G$ such that $ax = 
    b$ and $ya = b$.
\end{proposition}

\begin{enumerate}
    \item Prove Proposition~\ref{prop:semigroup2group}.
    
    \item Prove Proposition~\ref{prop:semigroup2group2}.
\end{enumerate}