Skip to main content
SpringerLink
Log in

Epimorphisms in varieties of subidempotent residuated structures

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

A commutative residuated lattice \({\varvec{A}}\) is said to be subidempotent if the lower bounds of its neutral element e are idempotent (in which case they naturally constitute a Brouwerian algebra \({\varvec{A}}^-\)). It is proved here that epimorphisms are surjective in a variety \({\mathsf {K}}\) of such algebras \({\varvec{A}}\) (with or without involution), provided that each finitely subdirectly irreducible algebra \({\varvec{B}}\in {\mathsf {K}}\) has two properties: (1) \({\varvec{B}}\) is generated by lower bounds of e, and (2) the poset of prime filters of \({\varvec{B}}^-\) has finite depth. Neither (1) nor (2) may be dropped. The proof adapts to the presence of bounds. The result generalizes some recent findings of G. Bezhanishvili and the first two authors concerning epimorphisms in varieties of Brouwerian algebras, Heyting algebras and Sugihara monoids, but its scope also encompasses a range of interesting varieties of De Morgan monoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, A.R., Belnap Jr., N.D.: Entailment: The Logic of Relevance and Necessity, vol. 1. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  2. Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)

    MATH  Google Scholar 

  3. Beth, E.W.: On Padoa’s method in the theory of definitions. Indag. Math. 15, 330–339 (1953)

    Article  MathSciNet  Google Scholar 

  4. Bezhanishvili, G., Moraschini, T., Raftery, J.G.: Epimorphisms in varieties of residuated structures. J. Algebra 492, 185–211 (2017)

    Article  MathSciNet  Google Scholar 

  5. Blok, W.J., Hoogland, E.: The Beth property in algebraic logic. Stud. Log. 83, 49–90 (2006)

    Article  MathSciNet  Google Scholar 

  6. Blok, W.J., Pigozzi, D.: A finite basis theorem for quasivarieties. Algebra Universalis 22, 1–13 (1986)

    Article  MathSciNet  Google Scholar 

  7. Blok, W.J., Pigozzi, D.: Algebraizable logics. Memoirs of the American Mathematical Society 396. American Mathematical Society, Providence (1989)

  8. Campercholi, M.A.: Dominions and primitive positive functions. J. Symbol. Logic 83, 40–54 (2018)

    Article  MathSciNet  Google Scholar 

  9. Czelakowski, J., Dziobiak, W.: Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Universalis 27, 128–149 (1990)

    Article  MathSciNet  Google Scholar 

  10. Dunn, J.M.: The algebra of intensional logics. Ph.D. Thesis, University of Pittsburgh (1966) (College Publications 2019)

  11. Dunn, J.M.: Algebraic completeness results for R-mingle and its extensions. J. Symbol. Log. 35, 1–13 (1970)

    Article  MathSciNet  Google Scholar 

  12. Esakia, L.L.: Topological Kripke models. Soviet Math. Dokl. 15, 147–151 (1974)

    MATH  Google Scholar 

  13. Esakia, L.L.: Heyting Algebras I. Duality Theory. Metsniereba Press, Tblisi (1985). (Russian)

    MATH  Google Scholar 

  14. Esakia, L.L., Grigolia, R.: The variety of Heyting algebras is balanced. In: XVI Soviet Algebraic Conference, Part II, Leningrad, pp. 37–38 (1981) (Russian)

  15. Fussner, W., Galatos, N.: Categories of models of R-mingle. Ann. Pure Appl. Log. 170, 1188–1242 (2019)

    Article  MathSciNet  Google Scholar 

  16. Gabbay, D.M., Maksimova, L.: Interpolation and Definability: Modal and Intuitionistic Logics. Oxford Logic Guides, vol. 46. Clarendon Press, Oxford (2005)

  17. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices. An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, New York (2007)

  18. Galatos, N., Přenosil, A.: Embedding bimonoids into involutive residuated lattices (manuscript in preparation)

  19. Galatos, N., Raftery, J.G.: Adding involution to residuated structures. Stud. Log. 77, 181–207 (2004)

    Article  MathSciNet  Google Scholar 

  20. Galatos, N., Raftery, J.G.: A category equivalence for odd Sugihara monoids and its applications. J. Pure Appl. Algebra 216, 2177–2192 (2012)

    Article  MathSciNet  Google Scholar 

  21. Galatos, N., Raftery, J.G.: Idempotent residuated structures: some category equivalences and their applications. Trans. Am. Math. Soc. 367, 3189–3223 (2015)

    Article  MathSciNet  Google Scholar 

  22. Hoogland, E.: Definability and interpolation: model-theoretic investigations. PhD. Thesis, Institute for Logic, Language and Computation, University of Amsterdam (2001)

  23. Isbell, J.R.: Epimorphisms and dominions. In: Eilenberg, S., et al (eds.) Proceedings of the Conference on Categorical Algebra (La Jolla, California, 1965), pp. 232–246. Springer, New York (1966)

  24. Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)

    Article  MathSciNet  Google Scholar 

  25. Jónsson, B.: Congruence distributive varieties. Math. Jpn. 42, 353–401 (1995)

    MathSciNet  MATH  Google Scholar 

  26. Kiss, E.W., Márki, L., Pröhle, P., Tholen, W.: Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity. Stud. Sci. Math. Hung. 18, 79–140 (1983)

    MathSciNet  MATH  Google Scholar 

  27. Kihara, H., Ono, H.: Interpolation properties, Beth definability properties and amalgamation properties for substructural logics. J. Log. Comput. 20, 823–875 (2010)

    Article  MathSciNet  Google Scholar 

  28. Kreisel, G.: Explicit definability in intuitionistic logic. J. Symbol. Log. 25, 389–390 (1960)

    Google Scholar 

  29. Kuznetsov, A.V.: On superintuitionistic logics. In: Proceedings of the International Congress of Mathematicians, Vancouver, BC, 1974, vol. 1, pp. 243–249. Canadian Mathematical Congress, Montreal (1975)

  30. Maksimova, L.L.: Craig’s interpolation theorem and amalgamated varieties of pseudoboolean algebras. Algebra Log. 16, 643–681 (1977)

    Article  Google Scholar 

  31. Maksimova, L.L.: Craig’s interpolation theorem and amalgamable manifolds. Soviet Math. Dokl. 18, 1511–1514 (1977)

    MATH  Google Scholar 

  32. Maksimova, L.L.: Intuitionistic logic and implicit definability. Ann. Pure Appl. Log. 105, 83–102 (2000)

    Article  MathSciNet  Google Scholar 

  33. Maksimova, L.L.: Implicit definability and positive logics. Algebra Log. 42, 37–53 (2003)

    Article  MathSciNet  Google Scholar 

  34. Meyer, R.K.: On conserving positive logics. Notre Dame J. Form. Log. 14, 224–236 (1973)

    Article  MathSciNet  Google Scholar 

  35. Meyer, R.K.: Improved decision procedures for pure relevant logic. In: Anderson, C.A., Zeleny, M. (eds.) Logic, Meaning and Computation, pp. 191–217. Kluwer, Dordrecht (2001)

    Chapter  Google Scholar 

  36. Meyer, R.K., Dunn, J.M., Leblanc, H.: Completeness of relevant quantification theories. Notre Dame J. Form. Log. 15, 97–121 (1974)

    Article  MathSciNet  Google Scholar 

  37. Moraschini, T., Raftery, J.G., Wannenburg, J.J.: Varieties of De Morgan monoids: minimality and irreducible algebras. J. Pure Appl. Algebra 223, 2780–2803 (2019)

    Article  MathSciNet  Google Scholar 

  38. Moraschini, T., Raftery, J.G., Wannenburg, J.J.: Varieties of De Morgan monoids: covers of atoms. Rev. Symbol. Log. 13, 338–374 (2020)

    Article  MathSciNet  Google Scholar 

  39. Moraschini, T., Raftery, J.G., Wannenburg, J.J.: Epimorphisms, definability and cardinalities. Stud. Log. 108, 255–275 (2020)

    Article  MathSciNet  Google Scholar 

  40. Moraschini, T., Wannenburg, J.J.: Epimorphism surjectivity in varieties of Heyting algebras. Ann. Pure Appl. Log. 171(9), 102824 (2020). https://doi.org/10.1016/j.apal.2020.102824

    Article  MathSciNet  MATH  Google Scholar 

  41. Olson, J.S., Raftery, J.G.: Positive Sugihara monoids. Algebra Universalis 57, 75–99 (2007)

    Article  MathSciNet  Google Scholar 

  42. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)

    Article  MathSciNet  Google Scholar 

  43. Ringel, C.M.: The intersection property of amalgamations. J. Pure Appl. Algebra 2, 341–342 (1972)

    Article  MathSciNet  Google Scholar 

  44. Slaney, J.K.: On the structure of De Morgan monoids with corollaries on relevant logic and theories. Notre Dame J. Form. Log. 30, 117–129 (1989)

    Article  MathSciNet  Google Scholar 

  45. Thistlewaite, P.B., McRobbie, M.A., Meyer, R.K.: Automated Theorem-proving in Non-classical Logics. Pitman, London (1988)

    MATH  Google Scholar 

  46. Urquhart, A.: Duality for algebras of relevant logics. Stud. Log. 56, 263–276 (1996)

    Article  MathSciNet  Google Scholar 

  47. Urquhart, A.: Beth’s definability theorem in relevant logics. In: Orlowska, E. (ed.) Logic at Work; Essays Dedicated to the Memory of Helena Rasiowa. Studies in Fuzziness and Soft Computing, vol. 24, pp. 229–234. Physica Verlag, New York (1999)

    Google Scholar 

  48. Wannenburg, J.J.: Semilinear De Morgan monoids and epimorphisms (manuscript in preparation)

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07, Prague 8, Czech Republic

    T. Moraschini

  2. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Private Bag X20,  Pretoria, 0028, South Africa

    J. G. Raftery & J. J. Wannenburg

  3. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    J. J. Wannenburg

Authors
  1. T. Moraschini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. G. Raftery
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. J. Wannenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. G. Raftery.

Additional information

Communicated by Presented by P. Jipsen.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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 CZ.02.2.69/0.0/0.0/17_050/0008361, OPVVV MŠMT, MSCA-IF Lidské zdroje v teoretické informatice. 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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moraschini, T., Raftery, J.G. & Wannenburg, J.J. Epimorphisms in varieties of subidempotent residuated structures. Algebra Univers. 82, 6 (2021). https://doi.org/10.1007/s00012-020-00694-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-020-00694-2

Keywords

Mathematics Subject Classification

Search

Navigation