# Differences

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action_lattices [2010/07/29 15:22] (current)
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+=====Action lattices=====
+
+
+Abbreviation: **ActLat**
+
+====Definition====
+An \emph{action lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,0,\cdot,1,^*,\backslash ,/\rangle$
+of type $\langle 2,2,0,2,0,1,2,2\rangle$ such that
+
+$\langle A,\vee,0,\cdot,1,^*\rangle$ is a [[Kleene algebra]]
+
+$\langle A,\vee,\wedge\rangle$ is a [[lattice]]
+
+$\backslash$ is the left residual of $\cdot$:  $y\leq x\backslash z\Longleftrightarrow xy\leq z$
+
+$/$ is the right residual of $\cdot$:  $x\leq z/y\Longleftrightarrow xy\leq z$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be action lattices. A morphism from $\mathbf{A}$
+to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$,
+$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash +y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(x^*)=h(x)^*$, $h(0)=0$, $h(1)=1$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  | |
+^[[Quasiequational theory]]  |undecidable |
+^[[First-order theory]]  |undecidable |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |yes |
+^[[Congruence modular]]  |yes |
+^[[Congruence n-permutable]]  |yes, $n=2$ |
+^[[Congruence regular]]  |no |
+^[[Congruence e-regular]]  |yes |
+^[[Congruence uniform]]  |no |
+^[[Congruence extension property]]  |no |
+^[[Definable principal congruences]]  |no |
+^[[Equationally def. pr. cong.]]  |no |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &3\\ +f(4)= &20\\ +f(5)= &149\\ +f(6)= &1488\\ +\end{array}$
+
+
+====Subclasses====
+[[Commutative action lattices]]
+
+
+====Superclasses====
+[[Action algebras]]
+
+[[Residuated lattices]]
+
+
+====References====
+
+[(Kozen1994>
+D. Kozen, \emph{On action algebras}, In J. van Eijck and A. Visser, editors,
+\emph{Logic and Information Flow}, MIT Press, 1994, 78--88
+see also http://www.cs.cornell.edu/kozen/papers/act.ps
+)]

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