Abstract
We expand the notion of characteristic formula to infinite finitely presented subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presented subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of countable algebras, while they are not axiomatizable by standard Yankov (Jankov) formulas. We also give the examples of intermediate logics that are not axiomatizable by characteristic formulas of infinite algebras. Further, using the Gödel–McKinsey–Tarski translation, we extend these results to the varieties of interior algebras and normal extensions of S4. For this, using Maksimova’s Translation Lemma, we show that a finite presentation of a given Heyting algebra can be extended to its modal span. So, Maksimova’s Lemma allows us to extend the properties established for the finitely presented Heyting algebras to interior algebras.
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Notes
- 1.
There are two transcriptions of the last name: ‘Yankov’, that is used by Zentralblatt, and ‘Jankov’ used by Mathematical Reviews. The latter transcription is used much more often even though the former is more accurate and, as V.A. Yankov mentioned to the author, the transcription “Yankov” is preferred by him.
- 2.
Independently the same observation was made in Butz (1998). In Blok and Pigozzi (1982) it was proven that in finitely approximated varieties with EDPC (and, as it is well known, the variety \(\mathscr {H}\) of all Heyting algebras is finitely approximated and enjoys EDPC) every s.i. finitely presented algebra is finite.
- 3.
The Heyting algebras finitely presented over variety of all Heyting algebras are studied in Butz (1998).
- 4.
The general fact that every finitely presented in a variety with EDPC s.i. algebra defines a splitting was observed in Blok and Pigozzi (1982).
- 5.
For definition see McKenzie (1972).
- 6.
The concatenations are often called ordered, linear or Troelstra sums. We are trying to avoid use of the term “sum” since it suggests some kind of commutativity which is not the case here.
- 7.
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Acknowledgements
Many thanks to A. Muravitsky, G. Bezhanishvili and F. Wolter for their fruitful discussions. The author is also indebted to L.L. Maksimova for pointing out that the Translation Lemma can be used in the proof of the Theorem 16.
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Citkin, A. (2018). Characteristic Formulas Over Intermediate Logics. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_5
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