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EXISTENTIALLY CLOSED BROUWERIAN SEMILATTICES

Published online by Cambridge University Press:  15 October 2019

LUCA CARAI  and
SILVIO GHILARDI
LUCA CARAI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY 1290 FRENGER MALL LAS CRUCES, NM 88003-8001, USA E-mail: lcarai@nmsu.edu
SILVIO GHILARDI
Affiliation:
DIPARTIMENTO DI MATEMATICA UNIVERSITÁ DEGLI STUDI DI MILANO VIA CESARE SALDINI 50 20133 MILANO, ITALY E-mail: silvio.ghilardi@unimi.it
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Abstract

The variety of Brouwerian semilattices is amalgamable and locally finite; hence, by well-known results [19], it has a model completion (whose models are the existentially closed structures). In this article, we supply a finite and rather simple axiomatization of the model completion.

Keywords

Primary 03G25Secondary 03C10, 06D20Brouwerian semilatticesexistentially closed structuresfinite duality
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1544 - 1575
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Albert, M. H. and Burris, S. N., Finite axiomatizations for existentially closed posets and semilattices. Order, vol. 3 (1986), no. 2, pp. 169–178.CrossRefGoogle Scholar
Bezhanishvili, G. and Jansana, R., Esakia style duality for implicative semilattices. Applied Categorical Structures, vol. 21 (2013), no. 2, pp. 181–208.CrossRefGoogle Scholar
Celani, S. A., Representation of Hilbert algebras and implicative semilattices. Central European Journal of Mathematics, vol. 1 (2003), no. 4, pp. 561–572.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J., Model Theory, third ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1990.Google Scholar
Darnière, L., On the model-completion of Heyting algebras, 2018, arXiv:1810.01704.Google Scholar
Darnière, L. and Junker, M., Model completion of varieties of co-Heyting algebras. Houston Journal of Mathematics, vol. 44 (2018), no. 1, pp. 49–82.Google Scholar
Diego, A., Sur les algèbres de Hilbert, Collection de Logique Mathématique, Séries A, Fasc. XXI, Gauthier-Villars; E. Nauwelaerts; Louvain, Paris, 1966. Translated from the Spanish by Luisa Iturrioz.Google Scholar
Ghilardi, S. and Van Gool, S. J., Monadic second order logic as the model companion of temporal logic, Proceedings of the 31st Annual ACM-IEEE Symposium on Logic in Computer Science (LICS 2016), ACM, New York, 2016, p. 10.Google Scholar
Ghilardi, S. and Van Gool, S. J., A model-theoretic characterization of monadic second order logic on infinite words, this Journal, vol. 82 (2017), no. 1, pp. 62–76.Google Scholar
Ghilardi, S. and Zawadowski, M., Model completions and r-Heyting categories. Annals of Pure and Applied Logic, vol. 88 (1997), no. 1, pp. 27–46.CrossRefGoogle Scholar
Ghilardi, S. and Zawadowski, M., Sheaves, Games, and Model Completions, Trends in Logic—Studia Logica Library, vol. 14, Kluwer Academic Publishers, Dordrecht, 2002.CrossRefGoogle Scholar
Köhler, P., Brouwerian semilattices. Transactions of the American Mathematical Society, vol. 268 (1981), no. 1, pp. 103–126.CrossRefGoogle Scholar
Lipparini, P., Locally finite theories with model companion. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti (8), vol. 72 (1982), no. 1, pp. 6–11 (1983).Google Scholar
McKay, C. G., The decidability of certain intermediate propositional logics, this Journal, vol. 33 (1968), pp. 258–264.Google Scholar
Nemitz, W. C., Implicative semi-lattices. Transactions of the American Mathematical Society, vol. 117 (1965), pp. 128–142.CrossRefGoogle Scholar
Renardel De Lavalette, G. R., Interpolation in fragments of intuitionistic propositional logic, this Journal, vol. 54 (1989), no. 4, pp. 1419–1430.Google Scholar
Shavrukov, V. Yu., Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 323 (1993), p. 82.Google Scholar
Vrancken-Mawet, L., Dualité pour les demi-lattis de Brouwer. Bulletin de la Société royale des sciences de Liège, vol. 55 (1986), no. 2, pp. 346–352.Google Scholar
Wheeler, W. H., Model-companions and definability in existentially complete structures. Israel Journal of Mathematics, vol. 25 (1976), no. 3–4, pp. 305–330.CrossRefGoogle Scholar