Mathematical Structures: Lattices

# Lattices

Difference (from prior major revision) (author diff)

 \href{http://math.chapman.edu/cgi-bin/structures?Lattices_mace}{Lattices (mace)}

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

\abbreviation{Lat}
\begin{definition}
A \emph{lattice} is a structure $\mathbf{L}=\left\langle L,\vee ,\wedge \right\rangle$, where $\vee$ and $\wedge$ are infix binary operations
called the \emph{join} and \emph{meet}, such that

$\vee ,\wedge$ are associative:  $(x\vee y)\vee z=x\vee (y\vee z)$,$\ (x\wedge y)\wedge z=x\wedge (y\wedge z)$

$\vee ,\wedge$ are commutative:  $x\vee y=y\vee x$, $x\wedge y=y\wedge x$

$\vee ,\wedge$ are absorbtive:  $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$.

Remark:
It follows that $\vee$ and $\wedge$ are idempotent: $x\vee x=x$, $x\wedge x=x$.

This definition shows that lattices form a variety.

A partial order $\leq$ is definable in any lattice by
$x\leq y\Longleftrightarrow x\wedge y=x$, or equivalently by
$x\leq y\Longleftrightarrow x\vee y=y$.

\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be 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{definition}
A \emph{lattice} is a structure $\mathbf{L}=\left\langle L,\vee ,\wedge \right\rangle$ of type $\left\langle 2,2\right\rangle$ such that

$\left\langle L,\vee \right\rangle$ and $\left\langle L,\wedge \right\rangle$ are
\href{Semilattices.pdf}{Semilattices}, and

$\vee ,\wedge$ are absorbtive:  $(x\vee y)\wedge x=x$, $(x\wedge y)\vee x=x$
\end{definition}

\begin{definition}
A \emph{lattice} is a structure $\mathbf{L}=\left\langle L,\leq \right\rangle$ that is a
\href{Partially_ordered_sets.pdf}{Partially ordered sets} in which all elements $x,y\in L$ have a

least upper bound:  $z=x\vee y\Longleftrightarrow x\leq z$, $y\leq z \mbox{and}\ \forall w\ (x\leq w$, $y\leq w\implies z\leq w)$

greatest lower bound:  $z=x\wedge y\Longleftrightarrow z\leq x$, $z\leq y \mbox{and}\ \forall w\ (w\leq x$, $w\leq y\implies w\leq z)$
\end{definition}

\begin{definition}
A \emph{lattice} is a structure $\mathbf{L}=\left\langle L,\vee ,\wedge ,\leq \right\rangle$ such that $\left\langle L,\leq \right\rangle$ is a
\href{Partially_ordered_sets.pdf}{Partially ordered sets} and the following quasiequations hold:

$\vee$-left:  $x\leq z$, $y\leq z\ \implies x\vee y\leq z$

$\vee$-right:  $z\leq x\implies z\leq x\vee y$, $\quad z\leq y\implies z\leq x\vee y$

$\wedge$-right:  $z\leq x$, $z\leq y\implies z\leq x\wedge y$

$\wedge$-left:  $x\leq z\implies x\wedge y\leq z$, $\quad y\leq z\implies x\wedge y\leq z$

Remark:
These quasiequations give a cut-free Gentzen system to decide the equational
theory of lattices.
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\left\langle P(S),\cup ,\cap ,\subseteq \right\rangle$, the collection of
subsets of a sets $S$, ordered by inclusion.

\href{http://math.chapman.edu/cgi-bin/structures?Lattices_mace}{Lattices (mace)}
\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 & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes

Nenosuke Funayama,Tadasi Nakayama,\emph{On the distributivity of a lattice of lattice-congruences},
\textbf{18}1942,553--554\href{http://www.ams.org/mathscinet-getitem?mr=7:236c}{MRreview}
\\\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

Bjarni J\'onsson,\emph{Universal relational systems},
Math. Scand.,
\textbf{4}1956,193--208\href{http://www.ams.org/mathscinet-getitem?mr=20 :3091}{MRreview}
\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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

Jobst Heitzig, J\"urgen Reinhold, \emph{Counting finite lattices},
Algebra Universalis,
\textbf{48}, 2002, 43--53\href{http://www.ams.org/mathscinet-getitem?mr=1 930 032}{MRreview}

\href{http://www.chapman.edu/~jipsen/gap/lat1_6.html}{Lattices of size 1 to 6}

\href{http://math.chapman.edu/cgi-bin/structures?Finite_lattices}{Finite lattices} of size $\le 7$

\href{Subdirectly_irreducible_lattices.pdf}{Subdirectly irreducible lattices} of size $\le 7$

\href{Lattices_not_in_some_subclasses.pdf}{Lattices not in some subclasses} of size $\le 7$
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Modular_lattices.pdf}{Modular lattices}

\href{Semidistributive_lattices.pdf}{Semidistributive lattices}

\href{Neardistributive_lattices.pdf}{Neardistributive lattices}

\href{Join-complete_lattices.pdf}{Join-complete lattices}

\href{Meet-complete_lattices.pdf}{Meet-complete lattices}

\end{subclasses}
\begin{superclasses}\

\href{Semilattices.pdf}{Semilattices}

\href{Weakly_associative_lattices.pdf}{Weakly associative lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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