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.
@article{bwmeta1.element.bwnjournal-article-fmv153i3p199bwm, author = {Saharon Shelah}, title = {$\sigma$-Entangled linear orders and narrowness of products of Boolean algebras}, journal = {Fundamenta Mathematicae}, volume = {154}, year = {1997}, pages = {199-275}, zbl = {0886.03035}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv153i3p199bwm} }
Shelah, Saharon. σ-Entangled linear orders and narrowness of products of Boolean algebras. Fundamenta Mathematicae, Tome 154 (1997) pp. 199-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv153i3p199bwm/
[00000] [ARSh 153] U. Abraham, M. Rubin and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of -dense real order types, Ann. Pure Appl. Logic 29 (1985), 123-206. | Zbl 0585.03019
[00001] [AbSh 106] U. Avraham [Abraham] and S. Shelah, Martin’s axiom does not imply that every two -dense sets of reals are isomorphic, Israel J. Math. 38 (1981), 161-176. | Zbl 0457.03048
[00002] [Bo] R. Bonnet, Sur les algèbres de Boole rigides, PhD thesis, Université Lyon 1, 1978.
[00003] [BoSh 210] R. Bonnet and S. Shelah, Narrow Boolean algebras, Ann. Pure Appl. Logic 28 (1985), 1-12.
[00004] [CK] C. C. Chang and H. J. Keisler, Model Theory, Stud. Logic Found. Math. 73, North-Holland, Amsterdam, 1973.
[00005] [EK] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275-285. | Zbl 0137.41904
[00006] [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. | Zbl 0788.03046
[00007] [MgSh433] M. Magidor and S. Shelah, Length of Boolean algebras and ultraproducts, preprint.
[00008] [M1] D. Monk, Cardinal Invariants of Boolean Algebras, Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 1990.
[00009] [M2] D. Monk, Cardinal Invariants of Boolean Algebras, Progr. Math. 142, Birkhäuser, Basel, 1996.
[00010] [RoSh 534] A. Rosłanowski and S. Shelah, Cardinal invariants of ultrapoducts of Boolean algebras, Fund. Math., to appear.
[00011] [RoSh 599] A. Rosłanowski and S. Shelah, More on cardinal functions on Boolean algebras, preprint.
[00012] [SaSh 553] O. Shafir and S. Shelah, More on entangled linear orders, preprint.
[00013] [Sh 620] S. Shelah, On measure algebra, in: Proc. Conf. Prague 1996, submitted.
[00014] [Sh 460] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel J. Math., submitted. | Zbl 0955.03054
[00015] [Sh 50] S. Shelah, Decomposing uncountable squares to countably many chains, J. Combin. Theory Ser. A 21 (1976), 110-114. | Zbl 0366.04009
[00016] [Sh 345] S. Shelah, Products of regular cardinals and cardinal invariants of products of Boolean algebras, Israel J. Math. 70 (1990), 129-187. | Zbl 0722.03038
[00017] [Sh 410] S. Shelah, More on Cardinal Arithmetic, Arch. Math. Logic 32 (1993), 399-428. | Zbl 0799.03052
[00018] [Sh 371] S. Shelah, Advanced: cofinalities of small reduced products, Chapter VIII of [Sh g].
[00019] [Sh 355] S. Shelah, has a Jonsson Algebra, Chapter II of [Sh g].
[00020] [Sh 345a] S. Shelah, Basic: Cofinalities of small reduced products, Chapter I of [Sh g].
[00021] [Sh:g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
[00022] [Sh 400] S. Shelah, Cardinal arithmetic, Chapter IX of [Sh g].
[00023] [Sh 345b] S. Shelah, Entangled orders and narrow Boolean algebras, Appendix 2 to [Sh g].
[00024] [Sh 405] S. Shelah, Vive la différence II. The Ax-Kochen isomorphism theorem, Israel J. Math. 85 (1994), 351-390. | Zbl 0812.03018
[00025] [Sh 430] S. Shelah, Further cardinal arithmetic, Israel J. Math. 95 (1996), 61-114. | Zbl 0864.03032
[00026] [To] S. Todorčević, Remarks on chain conditions in products, Compositio Math. 5 (1985), 295-302. | Zbl 0583.54003