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\section{Groups et al.}

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\subsection{Groupoids}

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\subsection{Semigroups}

Mnemonic:  Groups have 4 axioms; semigroups have 2.
Semigroups are halfway (\emph{semi}) to being groups.

\begin{tabular}{|l|l|}
\hline $G \circ G \to G$ & \\
\hline
\hline Closure  & If $a,b \in G$ then $a \circ b \in G$ \\
\hline \textbf{Associativity} & $a\circ(b \circ c) = (a \circ b) \circ c$ for
	all $a,b,c\in G$ \\
\hline
\end{tabular}

%% ----------------------------------------------------------------
\subsection{Monoids}

Mnemonic: the \emph{mon} prefix is reminiscent of the number 1, and 1 is often
an identity when the binary operation is multiplication.

\begin{tabular}{|l|l|}
\hline $G \circ G \to G$ & \\
\hline
\hline Closure  & If $a,b \in G$ then $a \circ b \in G$ \\
\hline Associativity & $a\circ(b \circ c) = (a \circ b) \circ c$ for all
	$a,b,c\in G$ \\
\hline \textbf{Identity} & $e \circ g = g \circ e = g$ for all $g \in G$ \\
\hline
\end{tabular}

%% ----------------------------------------------------------------
\subsection{Groups}

\begin{tabular}{|l|l|}
\hline $G \circ G \to G$ & \\
\hline
\hline Closure  & If $a,b \in G$ then $a \circ b \in G$ \\
\hline Associativity & $a\circ(b \circ c) = (a \circ b) \circ c$ for all
	$a,b,c\in G$ \\
\hline Identity & $e \circ g = g \circ e = g$ for all $g \in G$ \\
\hline \textbf{Inverses} & For all $g\in G$ there exists $g^{-1} \in G$ such
	that $g \circ g^{-1} = g^{-1} \circ g = e$ \\
\hline
\end{tabular}

%% ----------------------------------------------------------------
\subsection{Abelian groups}

\begin{tabular}{|l|l|}
\hline $G \circ G \to G$ & \\
\hline
\hline Closure  & If $a,b \in G$ then $a \circ b \in G$ \\
\hline Associativity & $a\circ(b \circ c) = (a \circ b) \circ c$ for all
	$a,b,c\in G$ \\
\hline Identity & $e \circ g = g \circ e = g$ for all $g \in G$ \\
\hline Inverses & For all $g\in G$ there exists $g^{-1} \in G$ such
	that $g \circ g^{-1} = g^{-1} \circ g = e$ \\
\hline \textbf{Commutavity} & $a\circ b = b \circ a$ for all $a, b \in G$ \\
\hline
\end{tabular}

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\section{Rings}

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\subsection{Rings}
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\subsection{Rings with unit}
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\subsection{Commutative rings}
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\subsection{Integral domains}
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\subsection{Division rings}
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\subsection{Fields}
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\subsection{The real numbers}

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\section{Modules and vector spaces}
