Brouwerian algebras

http://mathcs.chapman.edu/structuresold/files/Brouwerian_algebras.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 Brouwerian algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Brouwerian_algebras}{edit}

\abbreviation{BrA}

\begin{definition}
A \emph{Brouwerian algebra} is a structure $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that


$\langle A, \vee, \wedge, 1\rangle$ is a distributive lattice
with top


$\rightarrow$ gives the residual of $\wedge$:  $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Brouwerian algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
\end{morphisms}

\begin{definition}
A \emph{Brouwerian algebra} is a BL-algebra $\mathbf{A}=\langle A, \vee, \wedge, 1, \cdot, \rightarrow\rangle$ such that

$x\wedge y=x\cdot y$
\end{definition}

\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
Equational theory & decidable\\\hline
Quasiequational theory & decidable\\\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 & yes, $n=2$\\\hline
Congruence e-regular & yes, $e=1$\\\hline
Congruence uniform & no\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & yes\\\hline
Equationally def. pr. cong. & yes\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &3\\
f(6)= &5\\
f(7)= &8\\
f(8)= &15\\
f(9)= &26\\
f(10)= &47\\
f(11)= &82\\
f(12)= &151\\
f(13)= &269\\
f(14)= &494\\
f(15)= &891\\
f(16)= &1639\\
f(17)= &2978\\
f(18)= &5483\\
f(19)= &10006\\
f(20)= &18428\\
Values known up to size 49 [Erne, Heitzig, Reinhold (2002)]
\end{array}$
\end{finite_members}

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

\href{Generalized_Boolean_algebras.pdf}{Generalized Boolean algebras} 

\href{Heyting_algebras.pdf}{Heyting algebras} 

\end{subclasses}

\begin{superclasses}\ 

\href{Distributive_lattices.pdf}{Distributive lattices} 

\href{Basic_logic_algebras.pdf}{Basic logic algebras} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%