Journal of Symbolic Logic

Quasi-subtractive varieties

Tomasz Kowalski, Francesco Paoli, and Matthew Spinks

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety 𝕍 the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ-assertional logic of 𝕍. Moreover, if 𝕍 has a constant 1 in its type and is 1-subtractive, the deductive filters on A∈ 𝕍 of the 1-assertional logic of 𝕍 coincide with the 𝕍-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics.

Article information

Source
J. Symbolic Logic, Volume 76, Issue 4 (2011), 1261-1286.

Dates
First available in Project Euclid: 11 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1318338848

Digital Object Identifier
doi:10.2178/jsl/1318338848

Mathematical Reviews number (MathSciNet)
MR2895395

Zentralblatt MATH identifier
1254.03119

Citation

Kowalski, Tomasz; Paoli, Francesco; Spinks, Matthew. Quasi-subtractive varieties. J. Symbolic Logic 76 (2011), no. 4, 1261--1286. doi:10.2178/jsl/1318338848. https://projecteuclid.org/euclid.jsl/1318338848


Export citation

References

  • P. Aglianò and A. Ursini, On subtractive varieties II: General properties, Algebra Universalis, vol. 36 (1996), pp. 222–259.
  • ––––, On subtractive varieties III: From ideals to congruences, Algebra Universalis, vol. 37(1997), pp. 296–333.
  • ––––, On subtractive varieties IV: Definability of principal ideals, Algebra Universalis, vol. 38(1997), pp. 355–389.
  • M. Ardeshir and W. Ruitenburg, Basic propositional calculus I, Mathematical Logic Quarterly, vol. 44 (1998), pp. 317–343.
  • G. D. Barbour and J. G. Raftery, Quasivarieties of logic, regularity conditions and parameterized algebraization, Studia Logica, vol. 74 (2003), pp. 99–152.
  • W. J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the AMS, no. 396, American Mathematical Society, Providence, RI, 1989.
  • ––––, On the structure of varieties with equationally definable principal congruences IV, Algebra Universalis, vol. 31(1994), pp. 1–35.
  • W. J. Blok and J. G. Raftery, Ideals in quasivarieties of algebras, Models, algebras and proofs (X. Caicedo and C. H. Montenegro, editors), Leecture Notes in Pure and Applied Mathematics, Dekker, 1999, pp. 167–186.
  • ––––, Assertionally equivalent quasivarieties, International Journal of Algebra and Computation, vol. 18(2008), pp. 589–681.
  • F. Bou, F. Paoli, A. Ledda, and H. Freytes, On some properties of quasi-MV algebras and $\sqrt'$quasi-MV algebras. Part II, Soft Computing, vol. 12 (2008), pp. 341–352.
  • S. Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, no. 78, Springer, 1981.
  • I. Chajda, Congruence properties of algebras in nilpotent shifts of varieties, General algebra and discrete mathematics (K. Denecke and O. Lüders, editors), Heidermann, Berlin, 1995, pp. 35–46.
  • ––––, Normally presented varieties, Algebra Universalis, vol. 34(1995), pp. 327–335.
  • I. Chajda, R. Halas, J. Kühr, and A. Vanzurova, Normalization of MV algebras, Mathematica Bohemica, vol. 130 (2005), no. 3, pp. 283–300.
  • J. Czelakowski, Equivalential logics I, Studia Logica, vol. 45 (1981), pp. 227–236.
  • M. R. Darnel, Theory of lattice ordered groups, Dekker, New York, 1995.
  • J. Duda, Arithmeticity at $0$, Czechoslovak Mathematical Journal, vol. 37 (1987), pp. 197–206.
  • G. Epstein and A. Horn, Logics which are characterised by subresiduated lattices, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 22 (1976), pp. 199–210.
  • J. M. Font, R. Jansana, and D. Pigozzi, A survey of abstract algebraic logic, Studia Logica, vol. 74 (2003), pp. 13–97.
  • N. Galatos, P. Jipsen, T. Kowalski, and H. Ono, Residuated lattices: An algebraic glimpse on substructural logics, Logic and the Foundations of Mathematics, vol. 151, Elsevier, Amsterdam, 2007.
  • N. Galatos and C. Tsinakis, Generalized MV algebras, Journal of Algebra, vol. 283 (2005), no. 1, pp. 254–291.
  • E. Graczynska, On normal and regular identities, Algebra Universalis, vol. 27 (1990), pp. 387–397.
  • G. Grätzer, H. Lakser, and J. Płonka, Joins and direct products of equational classes, Canadian Mathematical Bulletin, vol. 12 (1969), pp. 741–744.
  • H. P. Gumm and A. Ursini, Ideals in universal algebra, Algebra Universalis, vol. 19 (1984), pp. 45–54.
  • B. Jónsson and C. Tsinakis, Products of classes of residuated structures, Studia Logica, vol. 77 (2004), pp. 267–292.
  • B. Klunder, Representable pseudo-interior algebras, Algebra Universalis, vol. 40 (1998), pp. 177–188.
  • T. Kowalski, Semisimplicity, EDPC and discriminator varieties of residuated lattices, Studia Logica, vol. 77 (2005), pp. 255–265.
  • T. Kowalski and F. Paoli, On some properties of quasi-MV algebras and $\sqrt'$quasi-MV algebras. Part III, Reports on Mathematical Logic, vol. 45 (2010), pp. 161–199.
  • ––––, Joins and subdirect products of varieties, Algebra Universalis, vol. 65(2011), pp. 371–391.
  • A. Ledda, M. Konig, F. Paoli, and R. Giuntini, MV algebras and quantum computation, Studia Logica, vol. 82 (2006), pp. 245–270.
  • A. I. Mal'cev, On the general theory of algebraic systems, Matematicheskii Sbornik (N.S.), vol. 35 (77) (1954), no. 1, pp. 3–20.
  • R. Mc Kenzie, An algebraic version of categorical equivalence for varieties and more general algebraic categories, Logic and algebra: Proceedings of the Magari conference (A. Ursini and P. Aglianó, editors), Dekker, New York, 1996, pp. 211–243.
  • R. Mc Kenzie, G. Mc Nulty, and W. Taylor, Algebras, lattices, varieties, vol. 1, Wadsworth & Brooks/Cole, Monterey, California, 1987.
  • L. I. Mel'nik, Nilpotent shifts of varieties, Mathematical Notes, vol. 14 (1973), pp. 692–696.
  • F. Paoli, A. Ledda, R. Giuntini, and H. Freytes, On some properties of quasi-MV algebras and $\sqrt'$quasi-MV algebras. Part I, Reports on Mathematical Logic, vol. 44 (2008), pp. 53–85.
  • G. Sambin and V. Vaccaro, Topology and duality in modal logic, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 249–296.
  • M. Spinks and R. Veroff, Constructive logic with strong negation is a substructural logic I, Studia Logica, vol. 88 (2008), pp. 325–348.
  • ––––, Constructive logic with strong negation is a substructural logic II, Studia Logica, vol. 89(2008), pp. 401–425.
  • A. Ursini, On subtractive varieties I, Algebra Universalis, vol. 31 (1994), pp. 204–222.
  • D. Vakarelov, Notes on N-lattices and constructive logic with strong negation, Studia Logica, vol. 36 (1977), pp. 109–125.
  • C. J. van Alten, An algebraic study of residuated ordered monoids and logics without exchange and contraction, Ph.D. thesis, University of Natal, 1998.
  • H. Werner, Discriminator algebras, Akademie Verlag, Berlin, 1978.