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  • Print publication year: 2019
  • Online publication date: March 2019

14 - Spectral Spaces via Model Theory

Summary

In this chapter we show how spectral spaces can be analyzed and constructed from a model-theoretic perspective. The reader is assumed to have seen firstorder predicate logic, including routine manipulations with formulas and structures; the precise setup is explained in Section 14.1. The central notion of the chapter is type space. In the classical meaning of the word, this is the spectrum (or “Stone-space”) S(T) of the Boolean algebra (called the Tarski–Lindenbaum algebra) of formulas, modulo equivalence with respect to a given theory T. Hence, in model theory, Boolean spaces are in the forefront.We will extend the setup by taking into account a given set Δ of formulas of interest, and adjust the type space to reflect this additional information. For example, in the language of rings one may want to focus on the set Δ of polynomial identities. Or, one considers instances of a given formula? (x, y) (i.e., we choose Δ to be the set of all formulas? (x, c), where c varies among the constants of a given language). One then essentially runs the classical construction of type spaces within Δ and obtains a set SΔ(T); see 14.2.1 and 14.2.4. In modern model theory, this construction is known under the name partial types and it is considered as a Boolean quotient space of S(T).

In 14.2.5 we show that SΔ(T) carries a natural, in general not Boolean, spectral topology; the open quasi-compact sets reflect lattice combinations of formulas from Δ. The space SΔ(T) is examined in Section 14.2. To see how the spectrum of a bounded distributive lattice, and thus any spectral space, fits into the model-theoretic setup, see 14.2.10; for the Zariski spectrum, see 14.2.9.

The second half of the chapter introduces a general method of attaching spectra to a given first-order structure A. One method is through expansions of A, see 14.2.12. The second, more general, method is exposed in Section 14.3 and defines spectra of A via morphisms into models of some axiomatizable class of structures (in a possibly larger language). For both methods the spectra are of the form SΔ(T) for suitably chosen data Δ and T. In 14.3.13 we show that the points of spectra attached to a structure A using the new methods admit an algebraic description in terms of expansions of A.

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