=====Implicative lattices=====

Abbreviation: **ImpLat**
====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 [[distributive lattices]]
$\to$ is an implication: 


$x\to(y\vee z) = (x\to y)\vee(x\to z)$


$x\to(y\wedge z) = (x\to y)\wedge(x\to z)$


$(x\vee y)\to z = (x\to z)\wedge(y\to z)$


$(x\wedge y)\to z = (x\to z)\vee(y\to z)$
==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: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$

Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures},
Order,
\textbf{15}1998,297--323[[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[[http://www.ams.org/mathscinet-getitem?mr=97g:06014|MRreview]]

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\
\end{array}$

====Subclasses====
[[Goedel algebras]] 

[[MV-algebras]] 

[[Lattice-ordered groups]] 

====Superclasses====
[[Distributive lattices]] 


====References====

[(Ln19xx>
)]