Mathematical Structures: History of Boolean semigroups

# History of Boolean semigroups

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 Revision 4 . . June 28, 2004 3:37 am by Jipsen Revision 3 . . (edit) April 30, 2003 6:37 pm by Peter Jipsen

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Changed: 1,125c1,120
 %%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 Boolean semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Boolean_semigroups}{edit} \abbreviation{BSgrp} \begin{definition} A \emph{Boolean semigroup} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\cdot\rangle$ such that $\langle A,\vee,0, \wedge,1,\neg\rangle$ is a \href{Boolean_algebras.pdf}{Boolean algebras} $\langle A,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups} $\cdot$ is \emph{join-preserving} in each argument: $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$ $\cdot$ is \emph{normal} in each argument: $0\cdot x=0 \mbox{ and } x\cdot 0=0$ Remark: \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$: $h(x\cdot y)=h(x)\cdot h(y)$ \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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &0\\ f(4)= &28\\ f(5)= &0\\ f(6)= &0\\ f(7)= &0\\ f(8)= &5457\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Boolean_monoids.pdf}{Boolean monoids} \href{Variety_generated_by_complex_algebras_of_semigroups.pdf}{Variety generated by complex algebras of semigroups} \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} %
 %%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 Boolean semigroups} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Boolean_semigroups}{edit} \abbreviation{BSgrp} \begin{definition} A \emph{Boolean semigroup} is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\cdot\rangle$ such that $\langle A,\vee,0, \wedge,1,\neg\rangle$ is a \href{Boolean_algebras.pdf}{Boolean algebra} $\langle A,\cdot\rangle$ is a \href{Semigroups.pdf}{semigroups} $\cdot$ is \emph{join-preserving} in each argument: $(x\vee y)\cdot z=(x\cdot z)\vee (y\cdot z) \mbox{ and } x\cdot (y\vee z)=(x\cdot y)\vee (x\cdot z)$ $\cdot$ is \emph{normal} in each argument: $0\cdot x=0 \mbox{ and } x\cdot 0=0$ \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be Boolean monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a Boolean homomorphism and preserves $\cdot$: $h(x\cdot y)=h(x)\cdot h(y)$ \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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &0\\ f(4)= &28\\ f(5)= &0\\ f(6)= &0\\ f(7)= &0\\ f(8)= &5457\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \begin{subclasses}\ \href{Boolean_monoids.pdf}{Boolean monoids} \href{Variety_generated_by_complex_algebras_of_semigroups.pdf}{Variety generated by complex algebras of semigroups} \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} %

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