Mathematical Structures: Implicative lattices

Implicative lattices

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 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Implicative lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Implicative_lattices}{edit} \abbreviation{ImpLat} \begin{definition} An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that $\langle A,\vee,\wedge\rangle$ is a \href{Distributive_lattices.pdf}{distributive lattices} $\to$ is an implication: $x\to(y\vee z) = (x\to y)\vee(x\to z)$ $x\to(y\wedge z) = (x\to y)\wedge(x\to z)$ $(x\vee y)\to z = (x\to z)\wedge(y\to z)$ $(x\wedge y)\to z = (x\to z)\vee(y\to z)$ \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$ \end{morphisms} Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures}, Order, \textbf{15}1998,297--323\href{"http://www.ams.org/mathscinet-getitem?mr=2001b:06013"}{MRreview} Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups}, Studia Logica, \textbf{56}1996,185--204\href{"http://www.ams.org/mathscinet-getitem?mr=97g:06014"}{MRreview} \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 & \\\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 & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Goedel_algebras.pdf}{Goedel algebras} \href{MV-algebras.pdf}{MV-algebras} \href{Lattice-ordered_groups.pdf}{Lattice-ordered groups} \end{subclasses} \begin{superclasses}\ \href{Distributive_lattices.pdf}{Distributive lattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Implicative lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Implicative_lattices}{edit} \abbreviation{ImpLat} \begin{definition} An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that $\langle A,\vee,\wedge\rangle$ is a \href{Distributive_lattices.pdf}{distributive lattices} $\to$ is an implication: $x\to(y\vee z) = (x\to y)\vee(x\to z)$ $x\to(y\wedge z) = (x\to y)\wedge(x\to z)$ $(x\vee y)\to z = (x\to z)\wedge(y\to z)$ $(x\wedge y)\to z = (x\to z)\vee(y\to z)$ \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$ \end{morphisms} Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures}, Order, \textbf{15}1998,297--323\href{"http://www.ams.org/mathscinet-getitem?mr=2001b:06013"}{MRreview} Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups}, Studia Logica, \textbf{56}1996,185--204\href{"http://www.ams.org/mathscinet-getitem?mr=97g:06014"}{MRreview} \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 & \\\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 & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Goedel_algebras.pdf}{Goedel algebras} \href{MV-algebras.pdf}{MV-algebras} \href{Lattice-ordered_groups.pdf}{Lattice-ordered groups} \end{subclasses} \begin{superclasses}\ \href{Distributive_lattices.pdf}{Distributive lattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Implicative_lattices.pdf
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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Implicative lattices}

\abbreviation{ImpLat}
\begin{definition}
An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that

$\langle A,\vee,\wedge\rangle$ is a \href{Distributive_lattices.pdf}{distributive lattices}
$\to$ is an implication:

$x\to(y\vee z) = (x\to y)\vee(x\to z)$

$x\to(y\wedge z) = (x\to y)\wedge(x\to z)$

$(x\vee y)\to z = (x\to z)\wedge(y\to z)$

$(x\wedge y)\to z = (x\to z)\vee(y\to z)$
\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism:

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

Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures},
Order,
\textbf{15}1998,297--323\href{"http://www.ams.org/mathscinet-getitem?mr=2001b:06013"}{MRreview}

Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups},
Studia Logica,
\textbf{56}1996,185--204\href{"http://www.ams.org/mathscinet-getitem?mr=97g:06014"}{MRreview}

\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 & \\\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 & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Goedel_algebras.pdf}{Goedel algebras}

\href{MV-algebras.pdf}{MV-algebras}

\href{Lattice-ordered_groups.pdf}{Lattice-ordered groups}

\end{subclasses}
\begin{superclasses}\

\href{Distributive_lattices.pdf}{Distributive lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%