\(\Gamma\)-deviation and localization (Q2762646)

If \(\Gamma\) is a set of linear order types (e.g., \(\Gamma= \{\omega\}\), \(\Gamma= \{\omega^*\}\)), then along the lines of other dimension-theories, the \(\Gamma\)-deviation or \(\Gamma\)-Krull-dimension of a poset \((P,\leq)\), denoted by \(k_\Gamma(P)\), is inductively defined with \(k_\Gamma (\text{antichain})=-1\) and \(k_\Gamma(P)= \alpha\) if for any \(\gamma\in \Gamma\) and any chain \(C\) of \(P\) of type \(\gamma\), there exists \(a<b\) in \(C\) such that \(b/a= \{x/a\leq x\leq b\}\), considered as an interval in \(P\), has \(k_\Gamma(b/a)= \beta\) for some \(\beta<\alpha\), while \(k_\Gamma(P)\) has not been previously defined. Various choices of \(\Gamma\) then produce variants of the ``categorical'' dimension idea expressed in the definition above. Furthermore, when \(L\) is a modular lattice and \(\sim\) is a congruence relation of \(L\), the problem and technique of lifting chains of type \(\gamma\) in \(L/ \sim\) to chains of type \(\gamma\) in \(L\) leads to a theory of localization of \(\Gamma\)-deviations which in turn provides techniques permitting proof of the claim that if \(\Gamma\) consists of indecomposable order types and if \(P\) is a strongly modular lattice then \(P\) fails to have a \(\Gamma\)-deviation iff \(P\) contains a \(\Gamma\)-dense set, leading to further consequences as well as partial converses.