Studia Logica

, Volume 107, Issue 2, pp 423–444 | Cite as

A Duality for Involutive Bisemilattices

  • Stefano BonzioEmail author
  • Andrea Loi
  • Luisa Peruzzi


We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.


Duality Involutive bisemilattice Stone space Płonka sum Paraconsistent weak Kleene 

Mathematics Subject Classification

Primary 08C20 Secondary 06E15 18A99 22A30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aqvist, L., Reflections on the logic of nonsense, Theoria 28(1):138–57, 1962.Google Scholar
  2. 2.
    Balbes, R., A representation theorem for distributive quasi-lattices, Fundamenta Mathematicae 68(2):207–214, 1970.CrossRefGoogle Scholar
  3. 3.
    Bonzio, S., Dualities for Płonka sums, Logica Universalis, forthcoming.
  4. 4.
    Bonzio, S., J. Gil-Férez, F. Paoli, and L. Peruzzi, On paraconsistent weak Kleene logic: axiomatization and algebraic analysis, Studia Logica 105(2):253–297, 2017.CrossRefGoogle Scholar
  5. 5.
    Bonzio, S., T. Moraschini, and M. Pra Baldi, Logics of left variable inclusion and Płonka sums of matrices. Submitted manuscript, 2018.
  6. 6.
    Bonzio, S., M. Pra Baldi, and D. Valota, Counting finite linearly ordered involutive bisemilattices. Submitted manuscript, 2018.Google Scholar
  7. 7.
    Brzozowski, J., De morgan bisemilattices, in 30th IEEE International Symposium on Multiple-Valued Logic, IEEE Press, 2000, pp. 23–25.Google Scholar
  8. 8.
    Ciuni, R., and M. Carrara, Characterizing logical consequence in paraconsistent weak Kleene, in L. Felline, A. Ledda, F. Paoli, and E. Rossanese, (eds.), New Developments in Logic and the Philosophy of Science, College, London, 2016.Google Scholar
  9. 9.
    Ciuni, R., T. Ferguson, and D. Szmuc, Logics based on linear orders of contaminating values. Submitted manuscript, 2017.Google Scholar
  10. 10.
    Clark, D., and B. Davey, Natural Dualities for the Working Algebraist. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1998.Google Scholar
  11. 11.
    Davey, B.A., Duality Theory on Ten Dollars a Day, Springer, Dordrecht, 1993, pp. 71–111.Google Scholar
  12. 12.
    Ferguson, T., A computational interpretation of conceptivism, Journal of Applied Non-Classical Logics 24(4):333–367, 2014.CrossRefGoogle Scholar
  13. 13.
    Finn, V.K., and R. Grigolia, Nonsense logics and their algebraic properties, Theoria 59(1–3):207–273, 1993.Google Scholar
  14. 14.
    Gierz, G., and A. Romanowska, Duality for distributive bisemilattices, Journal of the Australian Mathematical Society, A 51:247–275, 1991.CrossRefGoogle Scholar
  15. 15.
    Haimo, F., Some limits of Boolean algebras, Proceedings of the American Mathematical Society 2(4):566–576, 1951.CrossRefGoogle Scholar
  16. 16.
    Halldén, S., The Logic of Nonsense, Lundequista Bokhandeln, Uppsala, 1949.Google Scholar
  17. 17.
    Harding, J., and A.B. Romanowska, Varieties of birkhoff systems part I, Order 34(1):45–68, 2017.CrossRefGoogle Scholar
  18. 18.
    Harding, J., and A.B. Romanowska, Varieties of birkhoff systems part II, Order 34(1):69–89, 2017.CrossRefGoogle Scholar
  19. 19.
    Hofmann, K., M. Mislove, and A. Stralka, The Pontryagin Duality of Compact O-Dimensional Semilattices and its Applications. Lecture Notes in Mathematics. Springer, 1974.CrossRefGoogle Scholar
  20. 20.
    Kalman, J., Subdirect decomposition of distributive quasilattices, Fundamenta Mathematicae 2(71):161–163, 1971.CrossRefGoogle Scholar
  21. 21.
    Kleene, S., Introduction to Metamathematics, North Holland, Amsterdam, 1952.Google Scholar
  22. 22.
    Lakser, H., R. Padmanabhan, and C.R. Platt, Subdirect decomposition of Płonka sums, Duke Mathematical Journal 39:485–488, 1972.CrossRefGoogle Scholar
  23. 23.
    Ledda, A., Stone-type representations and dualities for varieties of bisemilattices, Studia Logica 106(2):417–448, 2018.CrossRefGoogle Scholar
  24. 24.
    Mardešić, S., and J. Segal, Shape Theory: The Inverse System Approach. North-Holland Mathematical Library, North-Holland, 1982.Google Scholar
  25. 25.
    McKenzie, R., and A. Romanowska, Varieties of distributive bisemilattices, Contributions to General Algebra 1:213–218, 1979.Google Scholar
  26. 26.
    Mobasher, B., D. Pigozzi, and G. Slutzki, Multi-valued logic programming semantics an algebraic approach, Theoretical Computer Science 171(1):77–109, 1997.CrossRefGoogle Scholar
  27. 27.
    Omori, H., Halldén’s logic of nonsense and its expansions in view of logics of formal inconsistency, in 27th International Workshop on Database and Expert Systems Applications, 2016, pp. 129–133.Google Scholar
  28. 28.
    Płonka, J., On a method of construction of abstract algebras, Fundamenta Mathematicae 61(2):183–189, 1967.CrossRefGoogle Scholar
  29. 29.
    Płonka, J., On distributive quasilattices, Fundamenta Mathematicae 60:191–200, 1967.CrossRefGoogle Scholar
  30. 30.
    Płonka, J., Some remarks on direct systems of algebras, Fundamenta Mathematicae 62(3):301–308, 1968.CrossRefGoogle Scholar
  31. 31.
    Płonka, J., On sums of direct systems of Boolean algebras, Colloquium Mathematicae 21:209–214, 1969.Google Scholar
  32. 32.
    Płonka, J., On the sum of a direct system of universal algebras with nullary polynomials, Algebra Universalis 19(2):197–207, 1984.CrossRefGoogle Scholar
  33. 33.
    Płonka, J., and A. Romanowska, Semilattice sums. Universal Algebra and Quasigroup Theory, 1992, pp. 123–158.Google Scholar
  34. 34.
    Priestley, H., Ordered topological spaces and the representation of distributive lattices, Proceedings of the London Mathematical Society 24:507–530, 1972.CrossRefGoogle Scholar
  35. 35.
    Priestley, H., Ordered sets and duality for distributive lattices, in M. Pouzet and D. Richard, (eds.), Orders: Description and Roles, vol. 99 of North-Holland Mathematics Studies, North-Holland, 1984, pp. 39–60.Google Scholar
  36. 36.
    Prior, A., Time and Modality, Oxford University Press, Oxford, 1957.Google Scholar
  37. 37.
    Romanowska, A., and J. Smith, Semilattice-based dualities, Studia Logica 56(1/2):225–261, 1996.CrossRefGoogle Scholar
  38. 38.
    Romanowska, A., and J. Smith, Duality for semilattice representations, Journal of Pure and Applied Algebra 115(3):289–308, 1997.CrossRefGoogle Scholar
  39. 39.
    Segerberg, K., A contribution to nonsense-logics, Theoria 31:199–217, 1964.CrossRefGoogle Scholar
  40. 40.
    Singer, I., and J. Thorpe, Lecture Notes on Elementary Topology and Geometry. Undergraduate Texts in Mathematics, Springer, New York, 1976.CrossRefGoogle Scholar
  41. 41.
    Stone, M., Applications of the theory of Boolean rings to general topology, Transactions of the American Mathematical Society 41:375–481, 1937.CrossRefGoogle Scholar
  42. 42.
    Szmuc, D., Defining LFIs and LFUs in extensions of infectious logics, Journal of Applied non Classical Logics 26(4):286–314, 2016.CrossRefGoogle Scholar
  43. 43.
    Szmuc, D., B.D. Re, and F. Pailos, Theories of truth based on four-valued infectious logics, Logic Journal of the IGPL, forthcoming.Google Scholar
  44. 44.
    Williamson, T., Vagueness, Routledge, London, 1994.Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Università Politecnica delle MarcheAnconaItaly
  2. 2.Università di CagliariCagliariItaly

Personalised recommendations