Abstract
There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of Saveliev (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017).
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Notes
Compare this with non-standard extensions, also used to prove assertions about the extended model, which are elementary; it is unclear, however, whether this technique produces as many results with no known alternative proofs as the technique based on ultrafilter extensions does. Interestingly, a recent paper [7] combines both techniques to obtain results in number theory.
In [25], it was erroneously stated that the set of right continuous maps forms a compact Hausdorff space w.r.t. the pointwise convergence topology; actually, the intended topology was a restricted pointwise convergence topology, as explained in details below.
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Acknowledgements
We would like to express our gratitude to Professors Robert I. Goldblatt and Neil Hindman who provided us some useful historical information. We are also indebted to two anonymous referees for some critical remarks and suggestions.
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Poliakov, N.L., Saveliev, D.I. On ultrafilter extensions of first-order models and ultrafilter interpretations. Arch. Math. Logic 60, 625–681 (2021). https://doi.org/10.1007/s00153-021-00783-6
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DOI: https://doi.org/10.1007/s00153-021-00783-6
Keywords
- Ultrafilter
- Ultrafilter quantifier
- Ultrafilter extension
- Ultrafilter interpretation
- First-order model
- Ultrafilter model
- Topological model
- Largest compactification
- Right continuous map
- Right open relation
- Right closed relation
- Regular closed set
- Limit of ultrafilter
- Restricted pointwise convergence topology
- Homomorphism
- Extension theorem