Mathematical Structures: Sequential algebras

# Sequential algebras

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 %%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 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} %
 %%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 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
%%run pdflatex

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\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 Sequential algebras}

\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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