History of Basic logic algebras

 
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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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