History of Sequential algebras

 
 \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 Sequential algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Sequential_algebras}{edit}
 
 \abbreviation{SeA}
 \begin{definition}
 A \emph{sequential algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
 \wedge,1,\neg,\circ,e,\triangleright,\triangleleft\rangle$ such that
 
 
 $\langle A,\vee,0,
 \wedge,1,\neg\rangle$ is a \href{Boolean_algebras.pdf}{Boolean algebras}
 
 
 $\langle A,\circ,e\rangle $ is a \href{Monoids.pdf}{monoids}
 
 
 $\triangleright$ is the \emph{right-conjugate} of $\circ$:  
 $(x\circ y)\wedge z=0 \iff (x\triangleright z)\wedge y=0$
 
 
 $\triangleleft$ is the \emph{left-conjugate} of $\circ$:  
 $(x\circ y)\wedge z=0 \iff (z\triangleleft y)\wedge x=0$
 
 
 $\triangleright,\triangleleft$ are \emph{balanced}:  
 $x\triangleright e=e\triangleleft x$
 
 
 $\circ$ is \emph{euclidean}:  
 $x\cdot(y\triangleleft z)\leq (x\cdot y)\triangleleft z$
 
 
 Remark: 
 
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be sequential algebras. 
 A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\circ$, $\triangleright$, $\triangleleft$, $e$:
 
 $h(x\circ y)=h(x)\circ h(y)$, $h(x\triangleright y)=h(x)\triangleright h(y)$, $h(x\triangleleft y)=h(x)\triangleleft h(y)$, $h(e)=e$
 \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 & undecidable\\\hline
 Quasiequational theory & undecidable\\\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 regular & yes\\\hline
 Congruence uniform & yes\\\hline
 Congruence extension property & yes\\\hline
 Definable principal congruences & yes\\\hline
 Equationally def. pr. cong. & yes\\\hline
 Discriminator variety & no\\\hline
 Amalgamation property & no\\\hline
 Strong amalgamation property & no\\\hline
 Epimorphisms are surjective & no\\\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{Relation_algebras.pdf}{Relation algebras} 
 
 \href{Representable_sequential_algebras.pdf}{Representable sequential algebras} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Distributive_residuated_lattices.pdf}{Distributive residuated lattices} 
 
 \href{Semiassociative_sequential_algebras.pdf}{Semiassociative sequential algebras} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
 %