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implicative_lattices [2010/07/29 15:46] (current)
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+=====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>
+)]