Mathematical Structures: History of Semilattices with zero

# History of Semilattices with zero

 %%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 Semilattices with zero} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Semilattices_with_zero}{edit} \abbreviation{Slat$_0$} \begin{definition} A \emph{semilattice with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle$ such that $\langle S,\cdot\rangle$ is a \href{Semilattices.pdf}{semilattices} $0$ is a zero for $\cdot$: $x\cdot 0=0$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with zero. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$ \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 & decidable in PTIME\\\hline Quasiequational theory & decidable\\\hline First-order theory & undecidable\\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & no\\\hline Congruence n-permutable & no\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline Congruence extension property & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Semilattices_with_identity_and_zero.pdf}{Semilattices with identity and zero} \end{subclasses} \begin{superclasses}\ \href{Semilattices.pdf}{Semilattices} \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 Semilattices with zero} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Semilattices_with_zero}{edit} \abbreviation{Slat$_0$} \begin{definition} A \emph{semilattice with zero} is a structure $\mathbf{S}=\langle S,\cdot,0\rangle$ of type $\langle 2,0\rangle$ such that $\langle S,\cdot\rangle$ is a \href{Semilattices.pdf}{semilattices} $0$ is a zero for $\cdot$: $x\cdot 0=0$ \end{definition} \begin{morphisms} Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with zero. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$, $h(0)=0$ \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 & decidable in PTIME\\\hline Quasiequational theory & decidable\\\hline First-order theory & undecidable\\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & no\\\hline Congruence modular & no\\\hline Congruence n-permutable & no\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline Congruence extension property & \\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & \\\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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Semilattices_with_identity_and_zero.pdf}{Semilattices with identity and zero} \end{subclasses} \begin{superclasses}\ \href{Semilattices.pdf}{Semilattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %