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Monadic Bounded Algebras

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Abstract

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA’s is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the representation of MBA’s as powerset algebras of certain directed graphs with a set of “marked” points.

It is shown that there are only countably many varieties of MBA’s, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable.

Finitely generated MBA’s are shown to be finite, with the free algebra on r generators having exactly \({{2^{{{3.2}^r}.2^{2^r - 1}}}}\) elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r.

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References

  1. Akishev, Galym, Monadic Bounded Algebras. PhD thesis, Victoria University of Wellington, 2009. http://hdl.handle.net/10063/915.

  2. Bass Hyman: ‘Finite monadic algebras’. Proceedings of the American Mathematical Society 9(2), 258–268 (1958)

    Google Scholar 

  3. Beeson, Michael J. Foundations of Constructive Mathematics, Springer-Verlag, 1985.

  4. Birkhoff, Garrett, Lattice Theory, American Mathematical Society, third edition, 1967.

  5. Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, 1981.

  6. Chagrov, Alexander, and Michael Zakharyaschev, Modal Logic, Oxford University Press, 1997.

  7. Fourman, M. P., and D. S. Scott, ‘Sheaves and logic’, in M. P. Fourman, C. J. Mulvey, and D. S. Scott (eds.), Applications of Sheaves, volume 753 of Lecture Notes in Mathematics, Springer-Verlag, 1979, pp. 302–401.

  8. Goldblatt, Robert, ‘Functional monadic bounded algebras’, Studia Logica 96 (1):41–48, 2010 (this issue).

    Google Scholar 

  9. Goldblatt, Robert, ‘Metamathematics of modal logic, parts I and II’, Reports on Mathematical Logic, vols. 6 and 7, 1976. Reprinted in [12], pp. 9–79.

  10. Goldblatt, Robert, Axiomatising the Logic of Computer Programming, volume 130 of Lecture Notes in Computer Science, Springer-Verlag, 1982.

  11. Goldblatt Robert: ‘Varieties of complex algebras’. Annals of Pure and Applied Logic 44, 173–242 (1989)

    Article  Google Scholar 

  12. Goldblatt, Robert, Mathematics of Modality, CSLI Lecture Notes No. 43. CSLI Publications, Stanford University, 1993.

  13. Goldblatt Robert: ‘Elementary generation and canonicity for varieties of Boolean algebras with operators’. Algebra Universalis 34, 551–607 (1995)

    Article  Google Scholar 

  14. Goldblatt, Robert, Topoi. The Categorial Analysis of Logic. Dover Publications, Inc., Mineola, New York, 2006.

  15. Goldblatt, Robert, Ian Hodkinson, and Yde Venema, ‘Erdős graphs resolve Fine’s canonicity problem’, The Bulletin of Symbolic Logic 10 (2):186–208, June 2004.

    Google Scholar 

  16. Halmos, Paul R., ‘Algebraic logic I. Monadic Boolean algebras’, Compositio Mathematica 12:217–249, 1955. Reprinted in [17].

    Google Scholar 

  17. Halmos, Paul R., Algebraic Logic, Chelsea, New York, 1962.

  18. Hughes, G. E., and M. J. Cresswell, A New Introduction to Modal Logic, Routledge, 1996.

  19. Hughes, G. E., and D. G. Londey, The Elements of Formal Logic, Methuen, 1965.

  20. Jónsson Bjarni, Alfred Tarski: ‘Boolean algebras with operators, part I’. American Journal of Mathematics 73, 891–939 (1951)

    Article  Google Scholar 

  21. Lemmon E.J.: ‘Algebraic semantics for modal logics II’. The Journal of Symbolic Logic 31, 191–218 (1966)

    Article  Google Scholar 

  22. Monk J.D.: ‘On equational classes of algebraic versions of logic I’. Mathematica Scandinavica 27, 53–71 (1970)

    Google Scholar 

  23. Scott, Dana, ‘Existence and description in formal logic’, in Ralph Schoenman (ed.), Bertrand Russell: Philosopher of the Century, George, Allen and Unwin, 1967, pp. 181–200.

  24. Segerberg, Krister, An Essay in Classical Modal Logic, volume 13 of Filosofiska Studier, Uppsala Universitet, 1971.

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  1. Victoria University of Wellington, Wellington, New Zealand

    Galym Akishev & Robert Goldblatt

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  1. Galym Akishev
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  2. Robert Goldblatt
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Correspondence to Robert Goldblatt.

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Akishev, G., Goldblatt, R. Monadic Bounded Algebras. Stud Logica 96, 1–40 (2010). https://doi.org/10.1007/s11225-010-9269-z

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