Mathematical Structures: Boolean modules over a relation algebra

# Boolean modules over a relation algebra

HomePage | RecentChanges | Login

http://mathcs.chapman.edu/structuresold/files/Boolean_modules_over_a_relation_algebra.pdf
%%run pdflatex

%


\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 Boolean modules over a relation algebra}

\abbreviation{BRMod}
\begin{definition}
A \emph{Boolean module over a \href{Relation_algebras.pdf}{relation algebra}} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,f_r\ (r\in R)\rangle$ such that

$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a \href{Boolean_algebras.pdf}{Boolean algebra}

$f_r$ is \emph{join-preserving}: $f_r(x\vee y)=f_r(x)\vee f_r(y)$

$f_{r\vee s}(x)=f_r(x)\vee f_s(x)$

$f_r(f_s(x))=f_{r\circ s}(x)$

$f_{1'}$ is the identity map:  $f_{1'}(x)=x$

$f_0(x)=0$

$f_{r^\smile}(\neg (f_r(x)))\le \neg x$

Remark: Assuming that $f_r$ is order-preserving, the last identity is equivalent to the condition that $f_{r^\smile}$ and $f_r$ are conjugate operators.
It follows that $f_r$ is \emph{normal}: $f_r(0)=0$.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean modules over a realtion algebra.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves all $f_r$:

$h(f_r(x))=f_r(h(x))$
\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 & \\\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 & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\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 $n$.

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

\href{One-element_algebras.pdf}{One-element algebras}

\end{subclasses}
\begin{superclasses}\

\href{Boolean_algebras_with_operators.pdf}{Boolean algebras with operators}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%

HomePage | RecentChanges | Login