Basic logic algebras

http://math.chapman.edu/structuresold/files/Basic_logic_algebras.pdf

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\begin{document}
\textbf{\Large Basic logic algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Basic_logic_algebras}{edit}

\abbreviation{BLA}

\begin{definition}
A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle $ such that

$\left\langle A,\vee ,0,\wedge ,1\right\rangle $ is a 
\href{Bounded_lattices.pdf}{bounded lattice}

$\left\langle A,\cdot ,1\right\rangle $ is a \href{Commutative_monoids.pdf}{commutative monoid}

$\rightarrow $ gives the residual of $\cdot $:  $x\cdot y\leq z\Longleftrightarrow y\leq x\rightarrow z$

linearity:  $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL:  $x\cdot(x\rightarrow y)=x\wedge y$

Remark: 
The BL identity implies that the lattice is distributive.
\end{definition}

\begin{definition}
A \emph{basic logic algebra} is a \href{FLe-algebras.pdf}{FLe-algebra} $\mathbf{A}=\left\langle
A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle $ such that

linearity:  $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL:  $x\cdot (x\rightarrow y)=x\wedge y$

Remark: 
The BL identity implies that the identity element $1$ is the top of the lattice.
\end{definition}

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

$h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge
y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot
y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
\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 & decidable\\\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 e-regular & yes, $e=1$\\\hline
Congruence uniform & no\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & \\\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 $n$.

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

The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.
\end{finite_members}

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

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

\href{Heyting_algebras.pdf}{Heyting algebras} 

\end{subclasses}

\begin{superclasses}\ 

\href{Generalized_basic_logic_algebras.pdf}{Generalized basic logic algebras} 

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

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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