Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T21:19:21.495Z Has data issue: false hasContentIssue false
  • English
  • Français

Chains and antichains in interval algebras

Published online by Cambridge University Press:  12 March 2014

M. Bekkali
M. Bekkali*
Affiliation:
P. O. Box 3454, Boulder, Colorado 80307
Get access Rights & Permissions [Opens in a new window]

Abstract

Let κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2ℵ0 = ℵω1 that there is a boolean algebra B of size ℵω1 such that length(B) = ℵω1 is not attained and the incomparability of B is less than ℵω1. Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].

Keywords

Tree algebrainterval algebraorder set
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 860 - 867
Copyright
Copyright © Association for Symbolic Logic 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baumgartner, J., Chains and antichains in P(ω), this Journal, vol. 45 (1990), pp. 85–92.Google Scholar
[2]Baumgartner, J. and Komjáth, P., Boolean algebras in which every chain and antichain is countable, Fundamenta Mathematicae,. vol. 111 (1981), pp. 125–133.CrossRefGoogle Scholar
[3]Bonnet, R., Sikaddour, H., and Rubin, M., On Boolean algebras with well-founded sets of generators, Transactions of the American Mathematical Society (to appear).Google Scholar
[4]Monk, J. D., Cardinal functions on boolean algebras, Birkhäuser, Basel, 1990.CrossRefGoogle Scholar
[5]Monk, J. D. and Bonnet, R. (editors), Handbook of Boolean algebras, North-Holland, Amsterdam, 1989.Google Scholar
[6]Minler, E. and Pouzet, M., On the width of ordered sets and Boolean algebras, Algebra Universalis, vol. 23 (1986), pp. 242–253.Google Scholar
[7]Rubin, M., A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness, Transactions of the American Mathematical Society, vol. 278 (1983), pp. 65–89.CrossRefGoogle Scholar
[8]Todorcevic, S., Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, Rhode Island, 1989, p. 55.CrossRefGoogle Scholar
[9]Todorcevic, S., Remarks on chain condition in products, Compositio Mathematica, vol. 55 (1985), pp. 295–302.Google Scholar