Mathematical Structures: History of Sequential algebras

# History of Sequential algebras

 %%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} %