Mathematical Structures: Modular lattices

# Modular lattices

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

\abbreviation{MLat}

\begin{definition}
A \emph{modular lattice} is a \href{Lattices.pdf}{lattice} $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

\emph{modular identity}:  $((x\wedge z) \vee y) \wedge z = (x\wedge z) \vee (y\wedge z)$
\end{definition}

\begin{definition}
A \emph{modular lattice} is a \href{Lattices.pdf}{lattice} $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

\emph{modular law}: $x\le z\implies (x\vee y) \wedge z\le x\vee (y\wedge z)$
\end{definition}

\begin{definition}
A \emph{modular lattice} is a lattice $\mathbf{L}=\langle L,\vee,\wedge\rangle$ such that $\mathbf{L}$ has no sublattice isomorphic
to the pentagon $\mathbf{N}_{5}$
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%<param name=showControls value='false'>
%param name=structure value='
%structure name="Lattice1" size="5">
%
%  <element id="0" x=" 0" y="0"/>
%  <element id="1" x="-.5" y=".5"/>
%  <element id="2" x="-.5" y="1.5"/>
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%'>
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\end{definition}

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

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

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$M_3$
%<applet codebase="/" code="Graph.class" width=100 height=100>
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%<param name=structure value='
%<structure name="Modular_lattice1" size="5">
%  <element id="0" x=" 0" y="0"/>
%  <element id="1" x="-1" y="1"/>
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is the smallest nondistributive modular lattice. By a result of \cite{Dedekind1900}
this lattice occurs as a sublattice of every nondistributive
modular lattice.
\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 & undecidable \cite{Freese1980} \cite{Herrmann1983}\\\hline
Quasiequational theory & undecidable \cite{Lipshitz1974}\\\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 & no\\\hline
Congruence regular & no\\\hline
Congruence uniform & no\\\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 & no\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &4\\ f(6)= &\\ f(7)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Distributive_lattices.pdf}{Distributive lattices}

\href{Complete_modular_lattices.pdf}{Complete modular lattices}

\end{subclasses}
\begin{superclasses}\

\href{Semimodular_lattices.pdf}{Semimodular lattices}

\href{Geometric_lattices.pdf}{Geometric lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Dedekind1900}
Richard Dedekind, \emph{\"Uber die von drei Moduln erzeugte Dualgruppe},
Math. Ann., \textbf{53}, 1900, 371--403

\bibitem{Freese1980}
Ralph Freese, \emph{Free modular lattices},
Trans. Amer. Math. Soc., \textbf{261}, 1980, 81--91 \href{http://www.ams.org/mathscinet-getitem?mr=81k:06010}{MRreview}

\bibitem{Herrmann1983}
Christian Herrmann, \emph{On the word problem for the modular lattice with four free generators},
Math. Ann., \textbf{265}, 1983, 513--527 \href{http://www.ams.org/mathscinet-getitem?mr=84m:06014}{MRreview}

\bibitem{Lipshitz1974}
L. Lipshitz, \emph{The undecidability of the word problems for projective geometries and modular lattices},
Trans. Amer. Math. Soc., \textbf{193}, 1974, 171--180 \href{http://www.ams.org/mathscinet-getitem?mr=51 :295}{MRreview}
\end{thebibliography}
\end{document}
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