History of Implicative lattices

 
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 \begin{document}
 \textbf{\Large Implicative lattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Implicative_lattices}{edit}
 
 \abbreviation{ImpLat}
 \begin{definition}
 An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that 
 
 $\langle A,\vee,\wedge\rangle$ is a \href{Distributive_lattices.pdf}{distributive lattices}
 $\to$ is an implication: 
 
 
 $x\to(y\vee z) = (x\to y)\vee(x\to z)$
 
 
 $x\to(y\wedge z) = (x\to y)\wedge(x\to z)$
 
 
 $(x\vee y)\to z = (x\to z)\wedge(y\to z)$
 
 
 $(x\wedge y)\to z = (x\to z)\vee(y\to z)$
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
 homomorphism: 
 
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$
 \end{morphisms}
 
 Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures},
 Order,
 \textbf{15}1998,297--323\href{"http://www.ams.org/mathscinet-getitem?mr=2001b:06013"}{MRreview}
 
 Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups},
 Studia Logica,
 \textbf{56}1996,185--204\href{"http://www.ams.org/mathscinet-getitem?mr=97g:06014"}{MRreview}
 
 \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 & \\\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 & \\\hline
 Congruence regular & \\\hline
 Congruence uniform & \\\hline
 Congruence extension property & \\\hline
 Definable principal congruences & \\\hline
 Equationally def. pr. cong. & \\\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 $n$.
 
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 \end{finite_members}
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 \begin{subclasses}\ 
 
 \href{Goedel_algebras.pdf}{Goedel algebras} 
 
 \href{MV-algebras.pdf}{MV-algebras} 
 
 \href{Lattice-ordered_groups.pdf}{Lattice-ordered groups} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Distributive_lattices.pdf}{Distributive lattices} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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