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Compact cylindric set algebras. (English) Zbl 0576.03041

The paper states some results on cylindric algebraic model theory without proofs and contains some remarks concerning the possible improvements of the reported results.
An algebra with a universe which is a subset of a power set is said to be compact if the intersection of any subset of its universe with the finite intersection property is not empty. This simple notion appears to be a suitable tool to connect various important notions in algebraic model theory. The compact elements of \(Gs_{\alpha}^{reg}\cap Lf_{\alpha}\) are related to the elementary classes of models. On the other hand, for the simple members of \(Gs_{\alpha}^{reg}\cap Lf_{\alpha}\), compactness turns out to be the algebraic version of the universality of models. Moreover, using the notion of compactness, the well-known connection between saturated and universal models can be formulated algebraically even for finite dimensions. The compact representability of Boolean algebras can be generalized for some classes of cylindric algebras as follows: Any member of the classes \(Lf_{\alpha}\), \(Gws_{\alpha}\), \(Gs_{\alpha}\), \(Gs_{\alpha}^{reg}\cap Lf_{\alpha}\), \(Cs_{\alpha}^{reg}\cap Lf_{\alpha}\) is isomorphic to a compact member of the class concerned.
A new related result not in the paper is a possible algebraic generalization of the model theoretical result according to which finite elementarily equivalent models are isomorphic: If \b{A}\(\in Gs_{\alpha}^{reg}\cap Lf_{\alpha}\) is such that every subbase of \b{A} has power \(<\alpha \cap \omega\), then the compactness of the zero- dimensional part of \b{A} implies that every isomorphism from \b{A} onto a member of \(Gs_{\alpha}^{reg}\) is a lower-base-isomorphism.

MSC:

03G15 Cylindric and polyadic algebras; relation algebras
03C50 Models with special properties (saturated, rigid, etc.)
03C52 Properties of classes of models
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