University of Illinois Urbana-Champaign

Precipitous Ideals on Cbas (Complete Boolean Algebras)

Kajitori, Kazuaki

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Description

Title
Precipitous Ideals on Cbas (Complete Boolean Algebras)
Author(s)
Kajitori, Kazuaki
Issue Date
1986
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • The subject of the thesis is precipitous ideals on complete Boolean algebras (abbr. cBas) which are generalizations of precipitous ideals on power sets introduced by Jech and Prikry.
  • Various results on the original notion are generalized to the case of cBas, some by using a new tool (the sum operation of ideals on cBas introduced by J. Takahashi). Among those are: consistency strength of the existence of a precipitous ideal on a cBa (being that of a measurable); precipitousness of a (kappa)-complete (kappa)('+)-saturated ideal on a cBa; a game-theoretic characterization of the precipitousness; preservation theorems for precipitousness under Boolean extension.
  • Some results cannot be generalized. For example, it is shown that there is no (kappa)('+)-saturated (kappa)-complete ((kappa)('+)-incomplete) ideal on a (kappa)-collapsing algebra.
  • Also, it is shown, in analogy to an algebraic characterization of a compact cardinal, that it is consistent (modulo a compact cardinal) that every (kappa)-complete ideal on a ((kappa),(INFIN))-distributive cBa has a (kappa)-complete precipitous extension.
Graduation Semester
1986
Type of Resource
text
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http://hdl.handle.net/2142/71243

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