Mathematical Structures: Implicative lattices

[Home]Implicative lattices

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%%run pdflatex

%

 
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 \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 
%%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}
%







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Last edited July 8, 2004 2:42 pm by Jipsen (diff)
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