Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2018 | 32 | 247--262
Tytuł artykułu

An Infinite Natural Product

Warianty tytułu
Języki publikacji
We study a countably infinite iteration of the natural product between ordinals.We present an "effective" way to compute this countable natural product; in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product; this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals. (original abstract)
Słowa kluczowe
Opis fizyczny
  • University of Rome Tor Vergata, Italy
  • Altman H., Intermediate arithmetic operations on ordinal numbers, MLQ Math. Log. Q. 63 (2017), no. 3-4, 228-242.
  • Bachmann H., Transfinite Zahlen. Zweite, neubearbeitete Auflage, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 1, Springer-Verlag, Berlin-New York, 1967.
  • Berline Ch., Lascar D., Superstable groups, Stability in model theory (Trento, 1984), Ann. Pure Appl. Logic 30 (1986), 1-43.
  • Blass A., Gurevich Y., Program termination and well partial orderings, ACM Trans. Comput. Log. 9 (2008), Art. 18, 26 pp.
  • Brookfield G., The length of Noetherian polynomial rings, Comm. Algebra 31 (2003), 5591-5607.
  • Carruth P.W., Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262-271.
  • Chatyrko V.A., Ordinal products of topological spaces, Fund. Math. 144 (1994), 95-117.
  • Conway J.H., On numbers and games, London Mathematical Society Monographs, No. 6, Academic Press, London-New York, 1976.
  • Ehrlich P., The rise of non-Archimedean mathematics and the roots of a misconception. I: The emergence of non-Archimedean systems of magnitudes, Arch. Hist. Exact Sci. 60 (2006), 1-121.
  • Harris D., Semirings and $T_1$ compactifications. I, Trans. Amer. Math. Soc. 188 (1974), 241-258.
  • Harzheim E., Ordered sets, Advances in Mathematics 7, Springer, New York, 2005.
  • Hausdorff F., Mengenlehre, Dritte Auflage, Walter de Gruyter & Co., Berlin-Leipzig, 1935.
  • de Jongh D.H.J., Parikh R., Well-partial orderings and hierarchies, Nederl. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39 (1977), 195-207.
  • Lipparini P., An infinite natural sum, MLQ Math. Log. Q. 62 (2016), 249-257.
  • Lipparini P., Some transfinite natural sums, Preprint 2016, arXiv:1509.04078v2.
  • Lipparini P., Some transfinite natural products, in preparation.
  • Matsuzaka K., On the definition of the product of ordinal numbers, Sûgaku 8 (1956/1957), 95-96 (in Japanese).
  • Milner E.C., Basic wqo- and bqo-theory, in: I. Rival, Graphs and order. The role of graphs in the theory of ordered sets and its applications. Proceedings of the NATO Advanced Study Institute held in Banff, Alta., May 18-31, 1984, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 147, D. Reidel Publishing Co., Dordrecht, 1985, pp. 487-502.
  • Sierpiński W., Sur les séries infinies de nombres ordinaux, Fund. Math. 36 (1949), 248-253.
  • Sierpiński W., Sur les produits infinis de nombres ordinaux, Soc. Sci. Lett. Varsovie. C. R. Cl. III. Sci. Math. Phys. 43 (1950), 20-24.
  • Sierpiński W., Cardinal and ordinal numbers, Second revised edition, Monografie Matematyczne Vol. 34, Państwowe Wydawnictwo Naukowe, Warsaw, 1965.
  • Simpson S.G., Ordinal numbers and the Hilbert basis theorem, J. Symbolic Logic 53 (1988), 961-974.
  • Toulmin G.H., Shuffling ordinals and transfinite dimension, Proc. London Math. Soc. 4 (1954), 177-195.
  • Väänänen J., Wang T., An Ehrenfeucht-Fraïssé game for $ℒ_{ω_1ω}$, MLQ Math. Log. Q. 59 (2013), 357-370.
  • Wang T., An Ehrenfeucht-Fraïssé game for $ℒ_{ω_1ω}$, MSc Thesis, Universiteit van Amsterdam, 2012.
  • Wolk E.S., Partially well ordered sets and partial ordinals, Fund. Math. 60 (1967) 175-186.
Typ dokumentu
Identyfikator YADDA

Zgłoszenie zostało wysłane

Zgłoszenie zostało wysłane

Musisz być zalogowany aby pisać komentarze.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.