Mathematical Structures: FL-algebras

# FL-algebras

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

Changed: 61,65c61
 Equational theory & decidable Hiroakira Ono,Yuichi Komori,Logics without the contraction rule, J. Symbolic Logic, 501985,169--201\href{"http://www.ams.org/mathscinet-getitem?mr=87a:03053"}{MRreview}\href{"http://www.emis.de/MATH-item?0583.03018"}{ZMATH} implementation
 Equational theory & decidable \cite{OK1985}

Changed: 121c117,124
 \bibitem{Ln19xx}
 \bibitem{OK1985} Hiroakira Ono, Yuichi Komori, Logics without the contraction rule, J. Symbolic Logic, 501985, 169--201 \href{http://www.ams.org/mathscinet-getitem?mr=87a:03053}{MRreview} \href{http://www.emis.de/MATH-item?0583.03018}{ZMATH} \href{http://www.chapman.edu/~jipsen/reslat/}{implementation}

http://mathcs.chapman.edu/structuresold/files/FL-algebras.pdf
%%run pdflatex

%


\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 FL-algebras}

\abbreviation{FL}
\begin{definition}
A \emph{full Lambek algebra}, or \emph{FL-algebra}, is a structure $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \backslash, /, 0\rangle$
of type $\langle 2,0,2,0,2,1,2,2\rangle$ such that

$\langle A, \vee, \wedge, \cdot, 1, \backslash, /\rangle$ is a
\href{Residuated lattices.pdf}{residuated lattice} and

$0$ is an additional constant (can denote any element).
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be FL-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$, $h(0)=0$
\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 \cite{OK1985}
\\\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 & no\\\hline
Congruence e-regular & yes\\\hline
Congruence uniform & no\\\hline
Congruence extension property & no\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &9\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{Bounded_residuated_lattices.pdf}{Bounded residuated lattices} subvariety

\href{FLe-algebras.pdf}{FLe-algebras} subvariety

\href{FLw-algebras.pdf}{FLw-algebras} subvariety

\href{FLc-algebras.pdf}{FLc-algebras} subvariety

\href{Distributive_FL-algebras.pdf}{Distributive FL-algebras} subvariety

\end{subclasses}

\begin{superclasses}\

\href{Residuated_lattices.pdf}{Residuated lattices} reduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{OK1985}
Hiroakira Ono, Yuichi Komori,
\emph{Logics without the contraction rule},
J. Symbolic Logic,
\textbf{50}1985, 169--201
\href{http://www.ams.org/mathscinet-getitem?mr=87a:03053}{MRreview}
\href{http://www.emis.de/MATH-item?0583.03018}{ZMATH}
\href{http://www.chapman.edu/~jipsen/reslat/}{implementation}

\end{thebibliography}

\end{document}
%