Abstract
A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz and the Maltsev hierarchies.
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References
Bentz, W., and L. Sequeira, Taylor’s modularity conjecture holds for linear idempotent varieties, Algebra Universalis 71(2): 101–102, 2014.
Blok, W. J., and D. Pigozzi, Protoalgebraic logics, Studia Logica 45: 337–369, 1986.
Blok, W. J., and D. Pigozzi, Algebraizable logics, volume 396 of Mem. Amer. Math. Soc. A.M.S., Providence, 1989.
Blok, W. J., and D. Pigozzi, Algebraic semantics for universal Horn logic without equality, in A. Romanowska, and J. D. H. Smith, (eds.), Universal Algebra and Quasigroup Theory, Heldermann, Berlin, 1992, pp. 1–56.
Blok, W. J., and J. G. Raftery, On congruence modularity in varieties of logic, Algebra Universalis 45(1): 15–21, 2001.
Blok, W. J., and J. G. Raftery, Assertionally equivalent quasivarieties, International Journal of Algebra and Computation 18(4): 589–681, 2008.
Blok, W. J., and J. Rebagliato, Algebraic semantics for deductive systems, Studia Logica, Special Issue on Abstract Algebraic Logic, Part II 74(5): 153–180, 2003.
Czelakowski, J., Protoalgebraic logics, volume 10 of Trends in Logic—Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
Czelakowski, J., The Suszko Operator. Part I, Studia Logica 74(5): 181–231, 2003.
Czelakowski, J., and R. Jansana, Weakly algebraizable logics, The Journal of Symbolic Logic 65(2): 641–668, 2000.
Font, J. M., Abstract Algebraic Logic—An Introductory Textbook, volume 60 of Studies in Logic—Mathematical Logic and Foundations, College Publications, London, 2016.
Font, J. M., and R. Jansana, A general algebraic semantics for sentential logics, volume 7 of Lecture Notes in Logic. A.S.L., second edition 2017 edition, 2009. First edition 1996. Electronic version freely available through Project Euclid at projecteuclid.org/euclid.lnl/1235416965.
Font, J. M., R. Jansana, and D. Pigozzi, A survey on abstract algebraic logic, Studia Logica, Special Issue on Abstract Algebraic Logic, Part II, 74(1–2): 13–97, 2003. With an “Update” in 91 (2009), 125–130.
García, O. C., and W. Taylor, The Lattice of Interpretability Types of Varieties, volume 50. Mem. Amer. Math. Soc., 1984.
Grätzer, G., Two Maläcev-Type Theorems in Universal Algebra, Journal of Combinatorial Theory 8: 334–342, 1970.
Herrmann, B., Characterizing equivalential and algebraizable logics by the Leibniz operator, Studia Logica 58: 305–323, 1997.
Hobby, D., and R. McKenzie, The structure of finite algebras, volume 76 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1988.
Jansana, R., and T. Moraschini, The Poset of All Logics I: Interpretations and Lattice Structure. Submitted manuscript. Available on ArXiv, 2019.
Jansana, R., and T. Moraschini, The Poset of All Logics II: Leibniz Classes and Hierarchy. Submitted manuscript. Available on ArXiv, 2019.
Jansana, R., and T. Moraschini, The Poset of All Logics IV: Irreducible Elements. Manuscript, 2020.
Kearnes, K. A., and E. W. Kiss, The shape of congruences lattices, volume 222 of Mem. Amer. Math. Soc. American Mathematical Society, 2013. Monograph.
Moraschini, T., A study of the truth predicates of matrix semantics, Review of Symbolic Logic 11(4): 780–804, 2018.
Moraschini, T., On the complexity of the Leibniz hierarchy, Annals of Pure and Applied Logic 170(7): 805–824, 2019.
Moraschini, T., and J. G. Raftery, On prevarieties of logic, Algebra Universalis 80(37): 11, 2019.
Neumann, W. D., Representing varieties of algebras by algebras, Journal of the Australian Mathematical Society 11: 1–8, 1970.
Neumann, W. D., On Mal’cev conditions, Journal of the Australian Mathematical Society 17: 376–384, 1974.
Opršal, J., Taylor’s modularity conjecture and related problems for idempotent varieties, Order 35(3): 433–460, 2018.
Pixley, A. F., Local mal’cev conditions, Canadian Mathematical Bulletin 15: 559–568, 1972.
Raftery, J. G., The equational definability of truth predicates, Reports on Mathematical Logic 41: 95–149, 2006.
Raftery, J. G., A perspective on the algebra of logic, Quaestiones Mathematicae 34: 275–325, 2011.
Taylor, W., Characterizing Mal’cev conditions, Algebra Universalis 3: 351–397, 1973.
Taylor, W., Varieties obeying homotopy laws, Canadian Journal of Mathematics 29: 498–527, 1977.
Tschantz, S., Congruence permutability is join prime, Unpublished manuscript, 1996.
Wille, R., Kongruenzklassengeometrien, Number 113 in Springer Lecture Notes. 1970.
Acknowledgements
Thanks are due to James G. Raftery for rising the question about whether the theory of the Maltsev and Leibniz hierarchy could be, to some extent, unified. In addition, we gratefully acknowledge the comments and suggestions of the anonymous referees, which helped to improve the presentation of the paper. This research was partially supported by the research Grant 2017 SGR 95 from the government of Catalonia and by the Research Project MTM2016-74892-P from the government of Spain, which include feder funds from the European Union. Furthermore, the second author was supported by the Grant CZ.02.2.69/0.0/0.0/17_050/0008361, OPVVV MŠMT, MSCA-IF Lidské zdroje v teoretické informatice, and by the Beatriz Galindo grant BEAGAL18/00040 of the Spanish Ministry of Science, Innovation and Universities.
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Jansana, R., Moraschini, T. The Poset of All Logics III: Finitely Presentable Logics. Stud Logica 109, 539–580 (2021). https://doi.org/10.1007/s11225-020-09916-z
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DOI: https://doi.org/10.1007/s11225-020-09916-z