More on uniform ultrafilters over a singular cardinal

Moti Gitik
2020 Fundamenta Mathematicae  
We present some new results related to the character of uniform ultrafilters over a singular cardinal and the ultrafilter number. 1. Introduction. An ultrafilter is one of the basic notions and tools in mathematics. Let us state a few well-known relevant definitions. An ultrafilter D over a cardinal κ is called uniform iff |A| = κ for every A ∈ D. Intuitively this means that D concentrates on sets of size κ and cannot be reduced to a smaller size. A subset W of D is called a basis of D iff for
more » ... very A ∈ D there is B ∈ W such that B ⊆ * A, where B ⊆ * A means that |B \ A| < κ. In other words, W generates D modulo the ideal of subsets of κ of cardinality < κ. The character of D is the smallest possible cardinality of its basis: ch(D) = min({|W | | W is a basis of D}). Finally, the spectrum and the ultrafilter number of κ are defined as follows: sp(κ) = {ch(D) | D is a uniform ultrafilter over κ} and u(κ) = min({ch(D) | D is a uniform ultrafilter over κ}). S. Garti and S. Shelah [6] initiated the study of the character and the ultrafilter number of uniform ultrafilters over a singular cardinal. It turns out that two basic topics of set theory-large cardinals and cardinal arithmeticplay a crucial role here. Many nice additional results were obtained in the paper [5] by S. Garti, M. Magidor and S. Shelah. This was extended by J. Cummings and C. Morgan [1] to singular cardinals of uncountable cofinality. In [4] , it was shown that already at κ = ℵ ω it is possible to have the ultrafilter number smaller than 2 κ . A model with a uniform ultrafilter over a singular cardinal κ which has a singular character < 2 κ was constructed in [7]. [1] c Instytut Matematyczny PAN, 2020 M. Gitik The purpose of the present paper is to continue the study of uniform ultrafilters over singular cardinals. More specifically, we address the question of Garti, Magidor and Shelah [5], who ask whether a regular cardinal in the interval [κ + , 2 κ ] can be omitted from the spectrum of κ, provided 2 κ > κ + . It is shown that this is possible for specific classes of ultrafilters-limits (i.e., those of the form F -lim α<cof(κ) U α ) or strongly uniform ultrafilters (where D is called strongly uniform iff there exists an increasing sequence τ = τ α | α < cof(κ) such that for every A ∈ D, the set {α < cof(κ) | |A ∩ τ α | = τ α } is unbounded in cof(κ)). The paper is organized as follows: In Section 1 some further basic notions and facts are stated. In Section 2, strongly uniform ultrafilters are introduced and results on their characters are proved. In Section 3, characters of limits are studied. Section 4 continues this using square principles and anti-large cardinals assumptions. In Section 5, we deal with a non-splitting number r(κ) and point out some connections to weak compactness. In Section 6, a construction of a model with u str (ℵ ω ) = ℵ ω+2 is presented. It is the only section that uses forcing. Finally, in Section 7, open questions are stated. 2. Some general observation. If U is an ultrafilter over a set X and W is an ultrafilter over a set Y , then W is below U in the Rudin-Keisler order, written U ≥ R-K
doi:10.4064/fm845-5-2020 fatcat:boxb6ar5knaqto4krk67fmmb2a