Mathematical Structures: Orthomodular lattices

# Orthomodular lattices

Difference (from prior major revision) (author diff)

Changed: 32c32
 An orthomodular lattice is an \href{Ortholattices.pdf}{ortholattices} $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that
 An orthomodular lattice is an \href{Ortholattices.pdf}{ortholattice} $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that

http://mathcs.chapman.edu/structuresold/files/Orthomodular_lattices.pdf
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\begin{document}
\textbf{\Large Orthomodular lattices}

\abbreviation{OMLat}
\begin{definition}
An \emph{orthomodular lattice} is an \href{Ortholattices.pdf}{ortholattice} $\mathbf{L}=\langle L,\vee,0,\wedge,1,'\rangle$ such that

the orthomodular law holds:  $((x\wedge y)\vee y')\wedge y=x\wedge y$
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x')=h(x)'$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\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 & \\\hline
Quasiequational theory & \\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\hline
Congruence extension property & no\\\hline
Definable principal congruences & no\\\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 $n$.

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

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\begin{subclasses}\

\href{Modular_ortholattices.pdf}{Modular ortholattices}

\end{subclasses}

\begin{superclasses}\

\href{Ortholattices.pdf}{Ortholattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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