Mathematical Structures: Bounded lattices

# Bounded lattices

Difference (from prior major revision) (author diff)

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

\abbreviation{BLat}

\begin{definition}
A \emph{bounded lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1\rangle$ such that

$\langle L,\vee,\wedge\rangle$ is a \href{Lattice.pdf}{lattice}

$0$ is the least element:  $0\leq x$

$1$ is the greatest element:  $x\leq 1$
\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be bounded lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a
homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$
\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 & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & undecidable\\\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 & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
\end{tabular}
\end{properties}
\end{table}

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

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &5\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &15\\ f(7)= &53\\ f(8)= &222\\ f(9)= &1078\\ f(10)= &5994\\ \end{array}$\qquad
$\begin{array}{lr} f(11)= &37622\\ f(12)= &262776\\ f(13)= &2018305\\ f(14)= &16873364\\ f(15)= &152233518\\ \end{array}$\qquad
$\begin{array}{lr} f(16)= &1471613387\\ f(17)= &15150569446\\ f(18)= &165269824761\\ f(19)= &\\ f(20)= &\\ \end{array}$

\cite{HeiRei2002}
\end{finite_members}

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

\href{Bounded_modular_lattices.pdf}{Bounded modular lattices}

\href{Complete_lattices.pdf}{Complete lattices}

\end{subclasses}

\begin{superclasses}\

\href{Lattices.pdf}{Lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{HeiRei2002}
Jobst Heitzig and J\"urgen Reinhold, \emph{Counting finite lattices},
Algebra Universalis,
\textbf{48}, 2002, 43--53 \href{http://www.ams.org/mathscinet-getitem?mr=2003h:05013}{MRreview}

\end{thebibliography}

\end{document}
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