Elements of Cylindric Algebraic Model Theory

According to J. Donald Monk, one of the authors of ‘Cylindric Algebras’, the basic monograph on algebraic logic, the fact that the sets of all

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$$ \phi ^\mathfrak{M} $$

’s consisting of sequences satisfying the first order formula φ in the model

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$$ \mathfrak{M} $$

constitutes the universe of a cylindric set algebra is ‘the main motivating force for […] the whole topic of algebraic logic.’ (cf. [Mon,00] p. 453). Therefore, the investigation of cylindric set algebras from the point of view of their close links to first order models has a distinguished role in algebraic logic. In the course of this investigation, the specific properties of models (e.g. universality, homogeneity, saturatedness) become algebraic ones, and the various connections between models correspond to different kinds of isomorphisms between the cylindric set algebras concerned (see e.g. [Hen-Mon-Tar,85] 4.3.68(7) and (10), [Hen-Mon-Tar,85] pp. 37 and 45, [Mon,00] Sections 5 and 6).