filtration
Let $M=(W,R,V)$ be a Kripke model for a modal logic $L$. Let $\mathrm{\Delta}$ be a set of wff’s. Define a binary relation^{} ${\sim}_{\mathrm{\Delta}}$ on $W$:
$$w{\sim}_{\mathrm{\Delta}}u\mathit{\hspace{1em}\hspace{1em}}\text{iff}\mathit{\hspace{1em}\hspace{1em}}{\vDash}_{w}A\text{iff}{\vDash}_{u}A\text{for any}A\in \mathrm{\Delta}.$$ 
Then ${\sim}_{\mathrm{\Delta}}$ is an equivalence relation^{} on $W$. Let ${W}^{\prime}$ be the set of equivalence classes^{} of ${\sim}_{\mathrm{\Delta}}$ on $W$. It is easy to see that if $\mathrm{\Delta}$ is finite, so is ${W}^{\prime}$. Next, let
$${V}^{\prime}(p):=\{[w]\in {W}^{\prime}\mid w\in V(p)\}.$$ 
Then ${V}^{\prime}$ is a welldefined function. We call a binary relation ${R}^{\prime}$ on ${W}^{\prime}$ a filtration^{} of $R$ if

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$wRu$ implies $[w]{R}^{\prime}[u]$

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$[w]{R}^{\prime}[u]$ implies that for any wff $A$ with $\mathrm{\square}A\in \mathrm{\Delta}$, if ${\vDash}_{w}\mathrm{\square}A$, then ${\vDash}_{u}A$.
The triple ${M}^{\prime}:=({W}^{\prime},{R}^{\prime},{V}^{\prime})$ is called a filtration of the model $M$.
Proposition 1.
(Filtration Lemma) Let $\mathrm{\Delta}$ be a set of wff’s closed under^{} the formation of subformulas: any subformula of any formula^{} in $\mathrm{\Delta}$ is again in $\mathrm{\Delta}$. Then
$${M}^{\prime}{\vDash}_{[w]}A\mathit{\hspace{1em}\hspace{1em}}\text{\mathit{i}\mathit{f}\mathit{f}}\mathit{\hspace{1em}\hspace{1em}}M{\vDash}_{w}A.$$ 
Title  filtration 

Canonical name  Filtration1 
Date of creation  20130322 19:35:39 
Last modified on  20130322 19:35:39 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03B45 
\@unrecurse 