nLab
maximal consistent

Given a normal modal logic, $\Lambda$, a set, $\Gamma$ of formulae is said to be $\Lambda$-consistent if $\neg (\Gamma {⊢}_{\Lambda }\perp )$, i.e., $\perp$ is not deducible from $\Gamma$.

A set, $\Gamma$, of formulae is said to be $\Lambda$-maximal if it is consistent and, for any $\varphi \in {ℒ}_{\omega }(n)$ either $\varphi \in \Gamma$ or $\neg \varphi \in \Gamma$.

Important

If $\Gamma$ is a $\Lambda$-maximal set of formulae, then within the Lindenbaum-Tarski algebra, ${𝔄}_{\omega }^{\Lambda }$, the set $x_{\Lambda }=\{\mid \mid \varphi \mid \mid \mid \varphi \in \Lambda \}$ is an ultrafilter.

Canonical frame

Let $S_{\omega }^{\Lambda }=\{\Gamma \mid \Gamma \mathrm{is}\Lambda -\mathrm{maximal}\}$, then $\Gamma \leftrightarrow x_{\Gamma }$ is a bijection between $S_{\omega }^{\Lambda }$ and the set of ultrafilters of ${𝔄}_{\omega }^{\Lambda }$.

This set forms the set of states / worlds for the canonical frame of $\Lambda$. The relations are given by

$$R_i\Gamma \Delta \mathrm{if},\mathrm{and}\mathrm{only}\mathrm{if},{\diamond }_i\Delta \subseteq \Gamma .$$R_i \Gamma\Delta if, and only if, \Diamond_i\Delta \subseteq \Gamma.