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Measures Induced by Units

  • Giovanni Panti (a1) and Davide Ravotti (a2)

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.

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[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. 187215.
[2] Beck, M. and Robins, S., Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, 2007.
[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. 517525.
[4] Beynon, W. M., Duality theorems for finitely generated vector lattices, Proceedings of the London Mathematical Society, vol. 31 (1975), no. 3, pp. 114128.
[5] Bigard, A., Keimel, K., and Wolfenstein, S., Groupes et anneaux réticulés, Lecture Notes in Mathematics, Volume 608, Springer, 1977.
[6] Blok, W. J. and Ferreirim, I. M. A., On the structure of hoops, Algebra Universalis, vol. 43 (2000), no. 2–3, pp. 233257.
[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.
[8] Cignoli, R., D'Ottaviano, I., and Mundici, D., Algebraic foundations of many-valued reasoning, Trends in logic, vol. 7, Kluwer, 2000.
[9] Dvurečenskij, A., Subdirectly irreducible state-morphism BL-algebras, Archive for Mathematical Logic, vol. 50 (2011), no. 1–2, pp. 145160.
[10] Engelking, R., Dimension theory, North-Holland, 1978.
[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. 531555.
[12] Ewald, G., Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer, 1996.
[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. 405441.
[14] Goodearl, K. R., Partially ordered abelian groups with interpolation, American Mathematical Society, Providence, RI, 1986.
[15] Hájek, P., Metamathematics of fuzzy logic, Trends in Logic, vol. 4, Kluwer, 1998.
[16] Hurewicz, W. and Wallman, H., Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941.
[17] Kokorin, A. I. and Kopytov, V. M., Fully ordered groups, Wiley, 1974.
[18] Kroupa, T., Every state on semisimple MV-algebra is integral, Fuzzy Sets and Systems, vol. 157 (2006), no. 20, pp. 27712782.
[19] Kuipers, L. and Niederreiter, H., Uniform distribution of sequences, Dover, New York, 2006, First published in 1974 by Wiley-Interscience.
[20] Marra, V., The Lebesgue state of a unital abelian lattice-ordered group. II, Journal of Group Theory, vol. 12 (2009), no. 6, pp. 911922.
[21] Metcalfe, G., Olivetti, N., and Gabbay, D., Proof theory for fuzzy logics, Applied Logic Series, vol. 36, Springer, 2009.
[22] Mundici, D., Averaging the truth-value in Łukasiewicz logic, Studia Logica, vol. 55 (1995), no. 1, pp. 113127.
[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. 537549.
[24] Mundici, D., Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic, Studia Logica Library, vol. 35, Springer, 2011.
[25] Murty, M. R., Problems in analytic number theory, Graduate Texts in Mathematics, vol. 206, Springer, 2001.
[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. 275289.
[27] Panti, G., Invariant measures in free MV-algebras, Communications in Algebra, vol. 36 (2008), no. 8, pp. 28492861.
[28] Panti, G., Denominator-preserving maps, Aequationes Mathematicae, vol. 84 (2012), no. 1–2, pp. 1325.
[29] Rourke, C. P. and Sanderson, B. J., Introduction topiecewise-linear topology, Springer, 1972.
[30] Rudin, W., Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.
[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.
[32] Stein, P., Classroom Notes: A Note on the Volume of a Simplex, The American Mathematical Monthly, vol. 73 (1966), no. 3, pp. 299301.
[33] Yosida, K., On the representation of the vector lattice, Proceedings of the Imperial Academy of Tokyo, vol. 18 (1942), pp. 339342.
[34] Ziegler, G. M., Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer, 1995.

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