Mathematical Structures: FLe-algebras

# FLe-algebras

http://mathcs.chapman.edu/structuresold/files/FLe-algebras.pdf
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\begin{document}
\textbf{\Large FLe-algebras}

\abbreviation{FL$_e$}
\begin{definition}
A \emph{full Lambek algebra with exchange}, or \emph{FLe-algebra}, is a \href{FL-algebras.pdf}{FL-algebras}
$\langle A, \vee, 0, \wedge, T, \cdot, 1, \backslash, /\rangle$ such that

$\cdot$ is commutative:  $x\cdot y=y\cdot x$

Remark:

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

$h(x\vee y)=h(x)\vee h(y)$, $h(\bot )=\bot$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(\top )=\top$,
$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(1)=1$
\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\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &3\\ f(4)= &16\\ f(5)= &100\\ f(6)= &794\\ \end{array}$
\end{finite_members}
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\begin{subclasses}\

\href{FLew-algebras.pdf}{FLew-algebras}

\href{Distributive_FLe-algebras.pdf}{Distributive FLe-algebras}

\end{subclasses}
\begin{superclasses}\

\href{Commutative_residuated_lattices.pdf}{Commutative residuated lattices}

\href{FL-algebras.pdf}{FL-algebras}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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