Abstract
The “representation problem” in abstract algebraic logic is that of finding necessary and sufficient conditions for a structure, on a well defined abstract framework, to have the following property: that for every structural closure operator on it, every structural embedding of the expanded lattice of its closed sets into that of the closed sets of another structural closure operator on another similar structure is induced by a structural transformer between the base structures. This question arose from Blok and Jónsson abstract analysis of one of Blok and Pigozzis’s characterizations of algebraizable logics. The problem, which was later on reformulated independently by Gil-Férez and by Galatos and Tsinakis, was solved by Galatos and Tsinakis in the more abstract framework of the category of modules over a complete residuated lattice, and by Galatos and Gil-Férez in the even more abstract setting of modules over a quantaloid. We solve the representation problem in Blok and Jónsson’s original context of M-sets, where M is a monoid, and characterise the corresponding M-sets both in categorical terms and in terms of their inner structure, using the notions of a graded M-set and a generalized variable introduced by Gil-Férez.
Similar content being viewed by others
References
Adámek, J., H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Number 17 of Reprints in Theory and Applications of Categories, 2006. Reprint of the 1990 original, Wiley, New York.
Blok W.J., Jónsson B.: Equivalence of consequence operations. Studia Logica 83(1–3), 91–110 (2006)
Blok, W. J., and D. Pigozzi, Algebrizable Logics, Vol. 396, Memoirs of the American Mathematical Society, Providence, January 1989.
Blok, W. J., and D. Pigozzi, Local deduction theorems in algebraic logic, in H. Andréka J. D. Monk, and I. Németi (eds.), Algebraic Logic, Vol. 54, Colloquia Mathematica Societatis János Bolyai, North-Holland, Amsterdam, 1991, pp. 75–109.
Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag Graduate Texts in Mathematics, New York, 1981.
Crown G. D.: Projectives and injectives in the category of complete lattices with residuated mappings. Mathematische Annalen 187, 295–299 (1970)
Font, J. M., and R. Jansana, A General Algebraic Semantics for Sentential Logics, Second revised ed., Vol. 7, Lecture Notes in Logic. Association for Symbolic Logic, 2009. Electronic version freely available through Project Euclid at http://projecteuclid.org/euclid.lnl/1235416965.
Font, J. M., R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, Studia Logica (Special issue on Abstract Algebraic Logic, Part II) 74:13–97, 2003. With an update in Vol. 91:125–130, 2009.
Galatos, N., and J. Gil-Férez, Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π-institutions, Journal of Pure and Applied Algebra (to appear).
Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Vol. 151, Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 2007.
Galatos N., Tsinakis C.: Equivalence of closure operators: an order-theoretic and categorical perspective. Journal of Symbolic Logic 74(3), 780–810 (2009)
Gil-Férez, J., Categorical Application to Abstract Algebraic Logic, Ph.D. Dissertation, University of Barcelona, December 2009. Chapters 2 and 5 published as [13] and [9] respectively.
Gil-Férez J.: Representations of structural closure operators. Archive for Mathematical Logic 50, 45–73 (2011)
Raftery J.: Correspondences between Gentzen and Hilbert systems. The Journal of Symbolic Logic 71, 903–957 (2006)
Rebagliato J., Verdú V.: On the algebraization of some Gentzen systems. Fundamenta Informaticae (Special issue on Algebraic Logic and its Applications) 18, 319–338 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Font, J.M., Moraschini, T. M-Sets and the Representation Problem. Stud Logica 103, 21–51 (2015). https://doi.org/10.1007/s11225-013-9536-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-013-9536-x