Mathematical Structures: Sequential algebras

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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
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\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}
%

http://mathcs.chapman.edu/structuresold/files/Sequential_algebras.pdf

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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 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}
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