Mundici: La Logica dei Poliedri

Mundici, Daniele:
La Logica dei Poliedri
Bollettino dell'Unione Matematica Italiana Serie 9 1 (2008), fasc. n.2, p. 455-474, (Italian)
pdf (487 Kb), djvu (225 Kb). | MR 2424304 | Zbl 1207.06010

Sunto

We describe the deep relationship between rational polyhedra, weighted abstract simplicial complexes, and finitely presented MV-algebras – the algebras of finitely axiomatizable theories in Łukasiewicz infinite-valued logic. Combining Alexander's classical stellar machinery with the solution, by Wklodarczyk and Morelli, of the weak Oda conjecture on toric varieties, we shall present several results involving classification, measurability, dissectability, and computability.

Referenze Bibliografiche

[1] J. W. ALEXANDER, The combinatorial theory of complexes, Annals of Mathematics, 31 (1930), 292-320. | fulltext (doi) | MR 1502943 | Zbl 56.0497.02

[2] E. R. BERLEKAMP, Block coding for the binary symmetric channel with noiseless, delayless feedback, In: Error-correcting Codes. (Mann, H. B., Ed.,) Wiley, New York (1968), 330-335. | MR 234756

[3] M. BUSANICHE - D. MUNDICI, Geometry of Robinson consistency in Łukasiewicz logic, Annals of Pure and Applied Logic, 147 (2007), 1-22. | fulltext (doi) | MR 2328196 | Zbl 1125.03016

[4] J. W. S. CASSELS, An introduction to the geometry of numbers, Springer, Berlin, 1959. | MR 157947 | Zbl 0086.26203

[5] A. V. CHERNAVSKI and V. P. LEKSINE, Unrecognizability of manifolds, Annals of Pure and Applied Logic, 141 (2006), 325-335. | fulltext (doi) | MR 2234702 | Zbl 1115.57014

[6] F. CICALESE - D. MUNDICI, Learning and the art of fault-tolerant guesswork, Handbook Chapter, In: Perspectives on Adaptivity and Learning, I. Stamatescu, et al., Eds., Springer (2003), 115-140.

[7] F. CICALESE - D. MUNDICI - U. VACCARO, Rota-Metropolis cubic logic and Ulam-Rényi games, In: Algebraic Combinatorics and Computer Science: a tribute to Gian-Carlo Rota (H. Crapo, D. Senato, Eds.), Springer-Verlag (2001), 197-244. | MR 1854480 | Zbl 0978.03045

[8] R. CIGNOLI - D. MUNDICI, Stone duality for Dedekind $\sigma$-complete $\ell$-groups with order-unit, Journal of Algebra, 302 (2006), 848-861. | fulltext (doi) | MR 2293786 | Zbl 1108.06011

[9] R. L. O. CIGNOLI - I. M. L. D'OTTAVIANO - D. MUNDICI, Algebraic foundations of many-valued reasoning, volume 7 of Trends in Logic, Kluwer Academic Publishers, Dordrecht (2000). | fulltext (doi) | MR 1786097

[10] R. L. O. CIGNOLI - E. J. DUBUC - D. MUNDICI, Extending Stone duality to multisets and locally finite MV-algebras, J. Pure and Applied Algebra, 189 (2004), 37-59. | fulltext (doi) | MR 2038562 | Zbl 1055.06004

[11] R. CIGNOLI - G. A. ELLIOTT - D. MUNDICI, Reconstructing $C^*$-algebras from their Murray von Neumann orders, Advances in Mathematics, 101 (1993), 166-179,. | fulltext (doi) | MR 1242603

[12] P. ERDÖS - P. M. GRUBER - J. HAMMER, Lattice points, Pitman Monographs and Surveys in Pure and Applied Mathematics 39, Longman and Wiley, New York, 1989. | MR 1003606

[13] G. EWALD, Combinatorial convexity and algebraic geometry, Springer-Verlag, New York, 1996. | fulltext (doi) | MR 1418400 | Zbl 0869.52001

[14] J. GISPERT - D. MUNDICI - A. TORRENS, Ultraproducts of $\mathbb{Z}$ with an application to many-valued logics, Journal of Algebra, 219 (1999), 214-233. | fulltext (doi) | MR 1707669 | Zbl 0937.06008

[15] F. HIRZEBRUCH, Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei complexen Veränderlichen, Math. Annalen, 126 (1953), 1-22. | fulltext EuDML | fulltext (doi) | MR 62842

[16] H. W. E. JUNG, Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen $x, y$ in der Umbegung einer Stelle $x=a$, $y=b$, J. reine angew. Math., 133 (1908), 289-314. | fulltext EuDML | fulltext (doi) | MR 1580742

[17] C. G. LEKKERKERKER, Geometry of numbers, Wolters-Noordhoff, Groningen and North-Holland, Amsterdam, 1969. | MR 271032

[18] W. B. R. LICKORISH, Simplicial moves on complexes and manifolds, Geometry and Topology Monographs, Vol. 2: Proceedings of the Kirbyfest (1999), 299-320. | fulltext (doi) | MR 1734414 | Zbl 0963.57013

[19] C. MANARA- V. MARRA - D. MUNDICI, Lattice-ordered Abelian groups and Schauder bases of regular fans, Transactions of the American Mathematical Society, 359 (2007), 1593-1604. | fulltext (doi) | MR 2272142 | Zbl 1118.06009

[20] YU. I. MANIN, A course in mathematical logic, Springer-Verlag, New York, 1977. | MR 457126 | Zbl 0383.03002

[21] V. MARRA - D. MUNDICI, The Lebesgue state of a unital abelian lattice-ordered group, Journal of Group Theory, 10 (2007), 655-684. | fulltext (doi) | MR 2352036 | Zbl 1136.06009

[22] P. MCMULLEN, Valuations and dissections, In: Handbook of Convex Geometry, Vol. 2. P. M. GRUBER - J. M. WILLS, Eds., Elsevier (1993), 933-988. | MR 1243000 | Zbl 0791.52014

[23] R. MORELLI, The birational geometry of toric varieties, Journal of Algebraic Geometry, 5 (1996), 751-782. | MR 1486987 | Zbl 0871.14041

[24] A. MOSTOWSKI, L'oeuvre scientifique de Jan Łukasiewicz dans le domaine de la logique mathématique, Fundamenta Mathematicae, 44 (1957), 1-11. | fulltext EuDML | fulltext (doi) | MR 89801

[25] D. MUNDICI, Interpretation of AF $C^*$-algebras in Łukasiewicz sentential calculus, Journal of Functional Analysis, 65 (1986), 15-63. | fulltext (doi) | MR 819173 | Zbl 0597.46059

[26] D. MUNDICI, Satisfiability in many-valued sentential logic is NP-complete, Theoretical Computer Science, 52 (1987), 145-153 | fulltext (doi) | MR 918116 | Zbl 0639.03042

[27] D. MUNDICI, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Advances in Mathematics, 68 (1988), 23-39. | fulltext (doi) | MR 931170

[28] D. MUNDICI, Logic of infinite quantum systems, International Journal of Theoretical Physics, 32 (1993), 1941-1955. | fulltext (doi) | MR 1255397 | Zbl 0799.03019

[29] D. MUNDICI, Łukasiewicz normal forms and toric desingularizations, In: Proceedings Logic Colloquium '93, W. Hodges et al., Editors, Oxford University Press, (1996), 401-423. | MR 1428014

[30] D. MUNDICI, Tensor Products and the Loomis-Sikorski theorem for MV-algebras, Advances in Applied Mathematics, 22 (1999), 227-248. | fulltext (doi) | MR 1659410 | Zbl 0926.06004

[31] D. MUNDICI, Simple Bratteli diagrams with a Gödel incomplete isomorphism problem, Transactions of the American Mathematical Society, 356 (2004), 1937-1955. | fulltext (doi) | MR 2031047 | Zbl 1042.46033

[32] D. MUNDICI, Representation of $\sigma$-complete MV-algebras and their associated Dedekind $\sigma$-complete $\ell$-groups, Contemporary Mathematics, 419 (2006), 219-230. | fulltext (doi) | MR 2279120 | Zbl 1119.06009

[33] D. MUNDICI, Bookmaking over infinite-valued events, International Journal of Approximate Reasoning, 43 (2006), 223-240. | fulltext (doi) | MR 2269599 | Zbl 1123.03011

[34] D. MUNDICI, The Haar theorem for lattice-ordered abelian groups with order-unit, Discrete and Continuous Dynamical Systems, 21 (2008). | fulltext (doi) | MR 2385705 | Zbl 1154.28007

[35] D. MUNDICI - G. PANTI, Extending addition in Elliott's local semigroup, Journal of Functional Analysis, 117 (1993), 461-471. | fulltext (doi) | MR 1244944 | Zbl 0799.46077

[36] D. MUNDICI - G. PANTI, Decidable and undecidable prime theories in infinite-valued logic, Annals of Pure and Applied Logic, 108 (2001), 269-278. | fulltext (doi) | MR 1819059 | Zbl 1130.03019

[37] D. MUNDICI - C. TSINAKIS, Gödel incompleteness in AF $C^*$-algebras, Forum Mathematicum. In stampa. | fulltext (doi) | MR 2479290

[38] M. H. A. NEWMAN, On the foundations of combinatory analysis situs, Proc. Royal Academy of Amsterdam, 29 (1926), 610-641. | Zbl 53.0547.02

[39] T. ODA, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Springer-Verlag, New York, 1988. | fulltext EuDML | MR 922894 | Zbl 0628.52002

[40] G. PANTI, Dynamical properties of logical substitutions, Discrete and Continuous Dynamical Systems, 15 (2006), 237-258. | fulltext (doi) | MR 2191395 | Zbl 1102.03024

[41] G. PANTI, Bernoulli automorphisms of finitely generated free MV-algebras, Journal of Pure and Applied Algebra, 208 (2007), 941-950. | fulltext (doi) | MR 2283436 | Zbl 1110.06014

[42] G. PANTI, Invariant measures in free MV-algebras, In stampa su Communications in Algebra. http://arxiv.org/abs/math/0508445v2. | fulltext (doi) | MR 2440284 | Zbl 1154.06008

[43] A. PELC, Searching games with errors: fifty years of coping with liars, Theoretical Computer Science, 270 (2002), 71-109. | fulltext (doi) | MR 1871067 | Zbl 0984.68041

[44] A. RÉNYI, Napló az információelméletröl, Gondolat, Budapest, 1976. (English translation: A Diary on Information Theory, J. Wiley and Sons, New York, 1984). | MR 772204

[45] B. RIECAN - D. MUNDICI, Probability on MV-algebras, In: E. Pap (ed.), Handbook of Measure Theory, Vol. II, North-Holland, Amsterdam, (2001), 869-909. | fulltext (doi) | MR 1954631

[46] C. P. ROURKE B. J. SANDERSON, Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972. | MR 350744 | Zbl 0254.57010

[47] Z. SEMADENI, Schauder bases in Banach spaces of continuous functions, Lecture Notes in Mathematics, Springer-Verlag, Berlin, volume 918, 1982. | MR 653986 | Zbl 0478.46014

[48] A. TARSKI - J. ŁUKASIEWICZ, Investigations into the Sentential Calculus, In: Logic, Semantics, Metamathematics, Oxford University Press, 1956, 38-59. Reprinted by Hackett Publishing Company, Indianapolis, 1983. | MR 736686

[49] M. TSUJII, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373. | fulltext (doi) | MR 1835391 | Zbl 0969.37012

[50] S. ULAM, Adventures of a Mathematician, Scribner's, New York, 1976. | MR 485098 | Zbl 0352.01009

[51] H. WEYL, The elementary theory of convex polyhedra, In: Contributions to the theory of games, Vol. I, Annals of Mathematics Studies 24, University Press, Princeton NJ, 1950, 3-18. | MR 38088

[52] J. WKLODARCZYK, Decompositions of birational toric maps in blow-ups and blow-downs. Transactions of the American Mathematical Society, 373-411 (1997), 349. | fulltext (doi) | MR 1370654

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche