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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
 +)]