Mathematical Structures: History of Join-semidistributive lattices

# History of Join-semidistributive lattices

HomePage | RecentChanges | Login

 Revision 3 . . July 8, 2004 2:45 pm by Jipsen Revision 2 . . (edit) April 19, 2003 11:36 am by Peter Jipsen

Difference (from prior major revision) (no other diffs)

Changed: 1,109c1,109
 %%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 Join-semidistributive lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Join-semidistributive_lattices}{edit} \abbreviation{JsdLat} \begin{definition} A \emph{join-semidistributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ that satisfies the join-semidistributive law SD$_{\vee}$: $x\vee y=x\vee z\implies x\vee y=x\vee(y\wedge z)$ \end{definition} \begin{morphisms} Let $\mathbf{L}$ and $\mathbf{M}$ be join-semidistributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} $D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that is split into two elements $d,d'$ using Alan Day's doubling construction. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & quasivariety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & undecidable\\\hline Congruence distributive & yes\\\hline Congruence modular & yes\\\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 & no\\\hline Strong amalgamation property & no\\\hline Epimorphisms are surjective & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Semidistributive_lattices.pdf}{Semidistributive lattices} \end{subclasses} \begin{superclasses}\ \href{Lattices.pdf}{Lattices} \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 Join-semidistributive lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Join-semidistributive_lattices}{edit} \abbreviation{JsdLat} \begin{definition} A \emph{join-semidistributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ that satisfies the join-semidistributive law SD$_{\vee}$: $x\vee y=x\vee z\implies x\vee y=x\vee(y\wedge z)$ \end{definition} \begin{morphisms} Let $\mathbf{L}$ and $\mathbf{M}$ be join-semidistributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} $D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that is split into two elements $d,d'$ using Alan Day's doubling construction. \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & quasivariety\\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & undecidable\\\hline Congruence distributive & yes\\\hline Congruence modular & yes\\\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 & no\\\hline Strong amalgamation property & no\\\hline Epimorphisms are surjective & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Semidistributive_lattices.pdf}{Semidistributive lattices} \end{subclasses} \begin{superclasses}\ \href{Lattices.pdf}{Lattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

HomePage | RecentChanges | Login
Search: