σ-Entangled linear orders and narrowness of products of Boolean algebras

• Autorzy: Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

We investigate σ-entangled linear orders and narrowness of Boolean algebras. We show existence of σ-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.

Kategorie tematyczne

• 03E10: Ordinal and cardinal numbers
• 06A07: Combinatorics of partially ordered sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1997
• Tom: 153
• Numer: 3
• Strony: 199-275

Daty

wydano

1997

otrzymano

1994-06-26

poprawiono

1996-12-17

Twórcy

autor

Saharon Shelah

• Institute of Mathematics, The Hebrew University, 91 904 Jerusalem, Israel
• Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854, U.S.A.

Bibliografia

• [ARSh 153] U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $ℵ_1$-dense real order types, Ann. Pure Appl. Logic 29 (1985), 123-206.
• [AbSh 106] U. Avraham [Abraham] and S. Shelah, Martin's axiom does not imply that every two $ℵ_1$-dense sets of reals are isomorphic, Israel J. Math. 38 (1981), 161-176.
• [Bo] R. Bonnet, Sur les algèbres de Boole rigides, PhD thesis, Université Lyon 1, 1978.
• [BoSh 210] R. Bonnet and S. Shelah, Narrow Boolean algebras, Ann. Pure Appl. Logic 28 (1985), 1-12.
• [CK] C. C. Chang and H. J. Keisler, Model Theory, Stud. Logic Found. Math. 73, North-Holland, Amsterdam, 1973.
• [EK] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275-285.
• [HLSh 162] B. Hart, C. Laflamme and S. Shelah, Models with second order properties, V: A General principle, Ann. Pure Appl. Logic 64 (1993), 169-194.
• [MgSh433] M. Magidor and S. Shelah, Length of Boolean algebras and ultraproducts, preprint.
• [M1] D. Monk, Cardinal Invariants of Boolean Algebras, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 1990.
• [M2] D. Monk, Cardinal Invariants of Boolean Algebras, Progr. Math. 142, Birkhäuser, Basel, 1996.
• [RoSh 534] A. Rosłanowski and S. Shelah, Cardinal invariants of ultrapoducts of Boolean algebras, Fund. Math., to appear.
• [RoSh 599] A. Rosłanowski and S. Shelah, More on cardinal functions on Boolean algebras, preprint.
• [SaSh 553] O. Shafir and S. Shelah, More on entangled linear orders, preprint.
• [Sh 620] S. Shelah, On measure algebra, in: Proc. Conf. Prague 1996, submitted.
• [Sh 460] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel J. Math., submitted.
• [Sh 50] S. Shelah, Decomposing uncountable squares to countably many chains, J. Combin. Theory Ser. A 21 (1976), 110-114.
• [Sh 345] S. Shelah, Products of regular cardinals and cardinal invariants of products of Boolean algebras, Israel J. Math. 70 (1990), 129-187.
• [Sh 410] S. Shelah, More on Cardinal Arithmetic, Arch. Math. Logic 32 (1993), 399-428.
• [Sh 371] S. Shelah, Advanced: cofinalities of small reduced products, Chapter VIII of [Sh g].
• [Sh 355] S. Shelah, $ℵ_{ω+1}$ has a Jonsson Algebra, Chapter II of [Sh g].
• [Sh 345a] S. Shelah, Basic: Cofinalities of small reduced products, Chapter I of [Sh g].
• [Sh:g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
• [Sh 400] S. Shelah, Cardinal arithmetic, Chapter IX of [Sh g].
• [Sh 345b] S. Shelah, Entangled orders and narrow Boolean algebras, Appendix 2 to [Sh g].
• [Sh 405] S. Shelah, Vive la différence II. The Ax-Kochen isomorphism theorem, Israel J. Math. 85 (1994), 351-390.
• [Sh 430] S. Shelah, Further cardinal arithmetic, Israel J. Math. 95 (1996), 61-114.
• [To] S. Todorčević, Remarks on chain conditions in products, Compositio Math. 5 (1985), 295-302.