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The reaping and splitting numbers of nice ideals

100%
Filipów R.
Colloquium Mathematicum
|
2014
|
tom 134
|
nr 2
179-192
EN
We examine the splitting number 𝔰(B) and the reaping number 𝔯(B) of quotient Boolean algebras B = 𝓟(ω)/ℐ where ℐ is an $F_{σ}$ ideal or an analytic P-ideal. For instance we prove that under Martin's Axiom 𝔰(𝓟(ω)/ℐ) = 𝔠 for all $F_{σ}$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin's Axiom 𝔯(𝓟(ω)/ℐ) = 𝔠 for all $F_{σ}$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications of these characteristics to the ideal convergence of sequences of real-valued functions defined on the reals.
2
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On the difference property of families of measurable functions

100%
Filipów R.
Colloquium Mathematicum
|
2003
|
tom 97
|
nr 2
169-180
EN
We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.
3
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On ideal equal convergence

64%
Filipów R., Staniszewski M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
896-910
EN
We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów R., Szuca P., Three kinds of convergence and the associated I-Baire classes, J. Math. Anal. Appl., 2012, 391(1), 1–9]. We also solve a few problems posed in the paper by Das, Dutta and Pal.
4
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When does the Katětov order imply that one ideal extends the other?

51%
Barbarski P., Filipów R., Mrożek N., Szuca P.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 1
91-102
EN
We consider the Katětov order between ideals of subsets of natural numbers ("$≤_{K}$") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals 𝓘 for which $𝓘 ≤_{K} 𝒥 ⇒ 𝓘 ⊑ 𝒥$ for every ideal 𝒥. We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).
5
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On some properties of Hamel bases and their applications to Marczewski measurable functions

51%
Dorais F., Filipów R., Natkaniec T.
Open Mathematics
|
2013
|
tom 11
|
nr 3
487-508
EN
We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.
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