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The linear refinement number and selection theory

100%

Machura M., Shelah S., Tsaban B.

Fundamenta Mathematicae

2016

tom 234

nr 1

15-40

The linear refinement number 𝔩𝔯 is the minimal cardinality of a centered family in $[ω]^{ω}$ such that no linearly ordered set in $([ω]^{ω},⊆ *)$ refines this family. The linear excluded middle number 𝔩𝔵 is a variation of 𝔩𝔯. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that 𝔩𝔯 = 𝔩𝔵 = 𝔡 in all models where the continuum is at most ℵ₂, and that the cofinality of 𝔩𝔯 is uncountable. Using the method of forcing, we show that 𝔩𝔯 and 𝔩𝔵 are not provably equal to 𝔡, and rule out several potential bounds on these numbers. Our results solve a number of open problems.

On the dimension of the space of ℝ-places of certain rational function fields

88%

Banakh T., Kholyavka Y., Potyatynyk O., Machura M., Kuhlmann K.

Open Mathematics

2014

tom 12

nr 8

1239-1248

We prove that for every n ∈ ℕ the space M(K(x 1, …, x n) of ℝ-places of the field K(x 1, …, x n) of rational functions of n variables with coefficients in a totally Archimedean field K has the topological covering dimension dimM(K(x 1, …, x n)) ≤ n. For n = 2 the space M(K(x 1, x 2)) has covering and integral dimensions dimM(K(x 1, x 2)) = dimℤ M(K(x 1, x 2)) = 2 and the cohomological dimension dimG M(K(x 1, x 2)) = 1 for any Abelian 2-divisible coefficient group G.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Open Mathematics

2 Machura M.

1 Banakh T.

1 Kholyavka Y.

1 Kuhlmann K.

1 Potyatynyk O.

1 Shelah S.

1 Tsaban B.

1 2016

1 2014

Wyniki wyszukiwania

The linear refinement number and selection theory

On the dimension of the space of ℝ-places of certain rational function fields