Mathematical Structures: Boolean groups

# Boolean groups

http://mathcs.chapman.edu/structuresold/files/Boolean_groups.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Boolean groups}

\abbreviation{BGrp}
\begin{definition}
A \emph{Boolean group} is a \href{Monoids.pdf}{monoid} $\mathbf{M}=\langle M, \cdot, e\rangle$ such that

every element has order $2$:  $x\cdot x=e$.
\end{definition}

\begin{morphisms}
Let $\mathbf{M}$ and $\mathbf{N}$ be Boolean groups. A morphism from $\mathbf{M}$
to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\langle \{0,1\},+ ,0\rangle$, the two-element group with addition-mod-2.
This algebra generates the variety of Boolean groups.
\end{example}
\end{examples}

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

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable in polynomial time\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & decidable\\\hline
Locally finite & yes\\\hline
Residual size & 2\\\hline
Congruence distributive & no\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &0\\ f(4)= &1\\ f(5)= &0\\ f(6)= &0\\ f(7)= &0\\ f(8)= &1\\ \end{array}$
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{Trivial_algebras.pdf}{Trivial algebras}

\end{subclasses}
\begin{superclasses}\

\href{Abelian_groups.pdf}{Abelian groups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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