Densities of Ultraproducts of Boolean Algebras
نویسندگان
چکیده
We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density dA of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra. In set theoretic topology, considerable effort has been put into the study of cardinal invariants of topological spaces, see e.g. [Ju1] and [Ho], [Ju2]. In Monk’s book [Mo], similarly a systematic study of cardinal invariants of Boolean algebras is undertaken; in particular, the behaviour of these invariants with respect to algebraic constructions like taking subalgebras, quotients etc. is investigated. One of these is the ultraproduct construction, well known from model theory; cf. [ChK]. Many questions on ultraproducts are highly dependent on set theory; among the more recent results are those in Shelah’ s pcf theory dealing with the possible cofinalities cf ( ∏ α<κ λα/D) where the λα are regular cardinals, hence well-ordered in a natural way, and the ultraproduct has the resulting linear order. Monk’s book contains a list of 66 problems, three of which are answered (consistently) in this paper. Problem 9. Does there exist a system (Ai)i∈I of infinite Boolean algebras and an ultrafilter F on I such that d( ∏
منابع مشابه
1 5 A pr 1 99 4 DENSITIES OF ULTRAPRODUCTS OF BOOLEAN ALGEBRAS
We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density dA of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra. In set theoretic topology, considerable effort has been put into the stu...
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