Atom-canonicity in algebraic logic in connection to omitting types in modal fragments of L_{ω, ω}

2016/08/10 by Ahmed, Tarek Sayed
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1608.03513

Abstract

Fix 2n, Nr_n\CA_m(⊆ CA_n) denotes the class of n-neat reducts of CA_m's. The existence of certain finite relation algebras and finite CA_n's lacking relativized complete representations is shown to imply that the omitting types theorem (OTT) fails for L_n with respect to clique guarded semantics (which is an equivalent formalism of its packed fragments), and for the multi-dimensional modal logic S5ⁿ. Several such relation and cylindric algebras are explicitly exhibited using rainbow constructions and Monk-like algebras. Certain CA_n constructed to show non-atom canonicity of the variety S\Nr_n\CA_n+3 are used to show that Vaught's theorem (VT) for L_{ω, ω}, looked upon as a special case of OTT for L_{ω, ω}, fails almost everywhere (a notion to be defined below) when restricted to L_n. That VT fails everywhere for L_n, which is stronger than failing almost everywhere as the name suggests, is reduced to the existence, for each n

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