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Measures Induced by Units

Published online by Cambridge University Press:  12 March 2014

Giovanni Panti  and
Davide Ravotti
Giovanni Panti
Affiliation:
Department of Mathematics, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: giovanni.panti@uniud.it
Davide Ravotti
Affiliation:
Department of Mathematics, University of Udine, Via Delle Scienze 206, 33100 Udine, Italy, E-mail: davide.ravotti@gmail.com
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Abstract

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop—equivalently, in the enveloping lattice-ordered abelian group—amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals—in this context usually called states—amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.

Keywords

03G2506F2011K06cancellative hooplattice-ordered abelian groupMV-algebrastrong unituniform distributionCesàro mean
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 886 - 910
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Badouel, E., Chenou, J., and Guillou, G., An axiomatization of the token game based on Petri algebras, Fundamenta Informaticae, vol. 77 (2007), no. 3, pp. 187–215.Google Scholar
[2] Beck, M. and Robins, S., Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, 2007.Google Scholar
[3] Beck, M., Sam, S. V., and Woods, K. M., Maximal periods of (Ehrhart) quasi-polynomials, Journal of Combinatorial Theory. Series A, vol. 115 (2008), no. 3, pp. 517–525.Google Scholar
[4] Beynon, W. M., Duality theorems for finitely generated vector lattices, Proceedings of the London Mathematical Society, vol. 31 (1975), no. 3, pp. 114–128.Google Scholar
[5] Bigard, A., Keimel, K., and Wolfenstein, S., Groupes et anneaux réticulés, Lecture Notes in Mathematics, Volume 608, Springer, 1977.Google Scholar
[6] Blok, W. J. and Ferreirim, I. M. A., On the structure of hoops, Algebra Universalis, vol. 43 (2000), no. 2–3, pp. 233–257.Google Scholar
[7] Bromwich, T. J., An introduction to the theory of infinite series, Macmillan and Co., 1908, Available at the Open Library, http://openlibrary.org/books/OL7073755M.Google Scholar
[8] Cignoli, R., D'Ottaviano, I., and Mundici, D., Algebraic foundations of many-valued reasoning, Trends in logic, vol. 7, Kluwer, 2000.Google Scholar
[9] Dvurečenskij, A., Subdirectly irreducible state-morphism BL-algebras, Archive for Mathematical Logic, vol. 50 (2011), no. 1–2, pp. 145–160.Google Scholar
[10] Engelking, R., Dimension theory, North-Holland, 1978.Google Scholar
[11] Esteva, F., Godo, L., Hájek, P., and Montagna, F., Hoops and fuzzy logic, Journal of Logic and Computation, vol. 13 (2003), no. 4, pp. 531–555.Google Scholar
[12] Ewald, G., Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer, 1996.Google Scholar
[13] Fedel, M., Keimel, K., Montagna, F., and Roth, W., Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic, Forum Mathematicum, vol. 25 (2013), no. 2, pp. 405–441.Google Scholar
[14] Goodearl, K. R., Partially ordered abelian groups with interpolation, American Mathematical Society, Providence, RI, 1986.Google Scholar
[15] Hájek, P., Metamathematics of fuzzy logic, Trends in Logic, vol. 4, Kluwer, 1998.Google Scholar
[16] Hurewicz, W. and Wallman, H., Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941.Google Scholar
[17] Kokorin, A. I. and Kopytov, V. M., Fully ordered groups, Wiley, 1974.Google Scholar
[18] Kroupa, T., Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems, vol. 157 (2006), no. 20, pp. 2771–2782.Google Scholar
[19] Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, Dover, New York, 2006, First published in 1974 by Wiley-Interscience.Google Scholar
[20] Marra, V., The Lebesgue state of a unital abelian lattice-ordered group. II, Journal of Group Theory, vol. 12 (2009), no. 6, pp. 911–922.CrossRefGoogle Scholar
[21] Metcalfe, G., Olivetti, N., and Gabbay, D., Proof theory for fuzzy logics, Applied Logic Series, vol. 36, Springer, 2009.Google Scholar
[22] Mundici, D., Averaging the truth-value in Łukasiewicz logic, Studia Logica, vol. 55 (1995), no. 1, pp. 113–127.Google Scholar
[23] Mundici, D., The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete and Continuous Dynamical Systems, vol. 21 (2008), no. 2, pp. 537–549.Google Scholar
[24] Mundici, D., Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic, Studia Logica Library, vol. 35, Springer, 2011.CrossRefGoogle Scholar
[25] Murty, M. R., Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer, 2001.CrossRefGoogle Scholar
[26] Panti, G., The automorphism group of falsum-free product logic, Algebraic and proof-theoretic aspects of non-classical logics (Aguzzoli, S. et al., editors), Lecture Notes in Artificial Intelligence, vol. 4460, Springer, 2007, pp. 275–289.Google Scholar
[27] Panti, G., Invariant measures in free MV-algebras, Communications in Algebra, vol. 36 (2008), no. 8, pp. 2849–2861.Google Scholar
[28] Panti, G., Denominator-preserving maps, Aequationes Mathematicae, vol. 84 (2012), no. 1–2, pp. 13–25.CrossRefGoogle Scholar
[29] Rourke, C. P. and Sanderson, B. J., Introduction topiecewise-linear topology, Springer, 1972.Google Scholar
[30] Rudin, W., Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.Google Scholar
[31] Stanley, R. P., Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, 1997, corrected reprint of the 1986 original.Google Scholar
[32] Stein, P., Classroom Notes: A Note on the Volume of a Simplex, The American Mathematical Monthly, vol. 73 (1966), no. 3, pp. 299–301.CrossRefGoogle Scholar
[33] Yosida, K., On the representation of the vector lattice, Proceedings of the Imperial Academy of Tokyo, vol. 18 (1942), pp. 339–342.Google Scholar
[34] Ziegler, G. M., Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer, 1995.Google Scholar