Abstract
We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most \(\mathfrak {s}\) non-logical symbols and an axiomatization requiring at most \(\mathfrak {m}\) variables, if the epimorphisms into structures with at most \(\mathfrak {m}+\mathfrak {s}+\aleph _0\) elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic \(\,\vdash \) with suitable infinitary definability properties of \(\,\vdash \), while not making the standard but awkward assumption that \(\,\vdash \) comes furnished with a proper class of variables.
Keywords
Epimorphism Prevariety Quasivariety Beth definability Algebraizable logic Equivalential logicPreview
Unable to display preview. Download preview PDF.
Notes
Acknowledgements
This work received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.
References
- 1.Adámek, J., How many variables does a quasivariety need? Algebra Universalis 27:44–48, 1990.CrossRefGoogle Scholar
- 2.Bacsich, P. D., Model theory of epimorphisms, Canad. Math. Bull. 17:471–477, 1974.CrossRefGoogle Scholar
- 3.Banaschewski, B., and H. Herrlich, Subcategories defined by implications, Houston J. Math. 2:149–171, 1976.Google Scholar
- 4.Bezhanishvili, G., T. Moraschini, and J. G. Raftery, Epimorphisms in varieties of residuated structures, J. Algebra 492:185–211, 2017.CrossRefGoogle Scholar
- 5.Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Phil. Soc. 29:433–454, 1935.CrossRefGoogle Scholar
- 6.Blok, W. J., and E. Hoogland, The Beth property in algebraic logic, Studia Logica 83:49–90, 2006.CrossRefGoogle Scholar
- 7.Blok, W. J., and B. Jónsson, Equivalence of consequence operations, Studia Logica 83:91–110, 2006.CrossRefGoogle Scholar
- 8.Blok, W. J., and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society 396, Amer. Math. Soc., Providence, 1989.Google Scholar
- 9.Blok, W. J., and D. Pigozzi, Algebraic semantics for universal Horn logic without equality, in J. D. H. Smith, and A. Romanowska (eds.), Universal Algebra and Quasigroup Theory, Heldermann Verlag, Berlin, 1992, pp. 1–56.Google Scholar
- 10.Budkin, A., Dominions in quasivarieties of universal algebras, Studia Logica 78:107–127, 2004.CrossRefGoogle Scholar
- 11.Budkin, A., Dominions of universal algebras and projective properties, Algebra and Logic 47:304–313, 2008.CrossRefGoogle Scholar
- 12.Campercholi, M. A., Dominions and primitive positive functions, J. Symbolic Logic 83:40–54, 2018.CrossRefGoogle Scholar
- 13.Czelakowski, J., Equivalential logics (I), and (II), Studia Logica 40:227–236, and 355–372, 1981.Google Scholar
- 14.Czelakowski, J., Protoalgebraic Logics, Kluwer, Dordrecht, 2001.CrossRefGoogle Scholar
- 15.Czelakowski, J., and D. Pigozzi, Amalgamation and interpolation in abstract algebraic logic, in X. Caicedo, and C. H. Montenegro (eds.), Models, Algebras and Proofs, Lecture Notes in Pure and Applied Mathematics, No. 203, Marcel Dekker, New York, 1999, pp. 187–265.Google Scholar
- 16.Font, J. M., Abstract Algebraic Logic – An Introductory Textbook, Studies in Logic 60, College Publications, London, 2016.Google Scholar
- 17.Font, J. M., R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, and Update, Studia Logica 74:13–97, 2003, and 91:125–130, 2009.Google Scholar
- 18.Freyd, P., Abelian categories, Harper and Row, New York, 1964.Google Scholar
- 19.Gabbay, D. M., and L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Oxford Logic Guides 46, Clarendon Press, Oxford, 2005.Google Scholar
- 20.Gorbunov, V. A., Algebraic Theory of Quasivarieties, Consultants Bureau, New York, 1998.Google Scholar
- 21.Grätzer, G., and H. Lakser, A note on the implicational class generated by a class of structures, Canad. Math. Bull. 16:603–605, 1973.CrossRefGoogle Scholar
- 22.Herrmann, B., Equivalential and algebraizable logics, Studia Logica 57:419–436, 1996.CrossRefGoogle Scholar
- 23.Herrmann, B., Characterizing equivalential and algebraizable logics by the Leibniz operator, Studia Logica 58:305–323, 1997.CrossRefGoogle Scholar
- 24.Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras, Part II, North-Holland, Amsterdam, 1985.Google Scholar
- 25.Higgins, P., Epimorphisms and amalgams, Colloquium Mathematicum 56:1–17, 1988.CrossRefGoogle Scholar
- 26.Hoogland, E., Algebraic characterizations of various Beth definability properties, Studia Logica 65:91–112, 2000.CrossRefGoogle Scholar
- 27.Hoogland, E., Definability and interpolation: model-theoretic investigations, PhD. Thesis, Institute for Logic, Language and Computation, University of Amsterdam, 2001.Google Scholar
- 28.Isbell, J. R., Epimorphisms and dominions, in S. Eilenberg, et al. (eds.), Proceedings of the Conference on Categorical Algebra (La Jolla, California, 1965), Springer, New York, 1966, pp. 232–246.Google Scholar
- 29.Kreisel, G., Explicit definability in intuitionistic logic, J. Symbolic Logic 25:389–390, 1960.CrossRefGoogle Scholar
- 30.Łoś, J., and R. Suszko, Remarks on sentential logics, Proc. Kon. Nederl. Akad. van Wetenschappen, Series A 61:177–183, 1958.Google Scholar
- 31.Maksimova, L. L., Intuitionistic logic and implicit definability, Ann. Pure Appl. Logic 105:83–102, 2000.CrossRefGoogle Scholar
- 32.Maksimova, L. L., Implicit definability and positive logics, Algebra and Logic 42:37–53, 2003.CrossRefGoogle Scholar
- 33.Maltsev, A. I., Several remarks on quasivarieties of algebraic systems (Russian), Algebra i Logika 5:3–9, 1966.Google Scholar
- 34.Prucnal, T., and A. Wroński, An algebraic characterization of the notion of structural completeness, Bull. Sect. Logic 3:30–33, 1974.Google Scholar
- 35.Raftery, J. G., Correspondences between Gentzen and Hilbert systems, J. Symbolic Logic 71:903–957, 2006.CrossRefGoogle Scholar
- 36.Raftery, J. G., A non-finitary sentential logic that is elementarily algebraizable, J. Logic Comput. 20:969–975, 2010.CrossRefGoogle Scholar
- 37.Wasserman, D., Epimorphisms and Dominions in Varieties of Lattices, PhD thesis, University of California at Berkeley, 2001.Google Scholar
- 38.Wójcicki, R., Theory of Logical Calculi, Kluwer, Dordrecht, 1988.CrossRefGoogle Scholar