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1
Content available Divisibility in β N and *N
100%
Šobot B.
Reports on Mathematical Logic
|
2019
|
tom Vol. 54
65--82
EN
The paper first covers several properties of the extension of the divisibility relation to a set ∗ N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone-Čech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible.
2
Content available remote Bipartite pseudo-BL algebras
100%
Walendziak A. , Wojciechowska-Rysiawa M.
Demonstratio Mathematica
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2010
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tom Vol. 43, nr 3
487-496
EN
The class of bipartite pseudo-BL algebras (denoted by BP) and the class of strongly bipartite pseudo-BL algebras (denoted by BP0) are investigated. We prove that the class BP0 is a variety and show that BP is closed under subalgebras and arbitrary direct products but it is not a variety. We also study connections between bipartite pseudo-BL algebras and other classes of pseudo-BL algebras.
3
Content available remote Ultrafilter na lądzie i morzu
100%
Pecura R.
Pneumatyka
|
2002
|
tom nr 2
22-23
PL
Dobrze znana użytkownikom sprężonego powietrza firma ultrafilter działa nie tylko na lądzie. Urządzenia uzdatniające sprężone powietrze produkowane i serwisowane przez ultrafilter z powodzeniem pracują również poza stałym lądem. W styczniu tego roku zostały zainstalowane i uruchomione dwa zespoły filtracji i osuszania sprężonego powietrza na Morzu Bałtyckim na platformie wiertniczej należącej do Petrobaltic.
4
Content available remote Divisibility in the Stone-Čech compactification
100%
Šobot B.
Reports on Mathematical Logic
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2015
|
tom Vol. 50
53--66
EN
After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Čech compactification βN, we establish some results about one of such extensions. This provides us with one possible divisibility relation on βN, │~, and we introduce a few more, defined in a natural way. For some of them we find equivalent conditions for divisibility. Finally, we mention a few facts about prime and irreducible elements of (βN, ·). The motivation behind all this is to try to translate problems in elementary number theory into βN.
5
Content available remote Structure spaces for rings of continuous functions with applications to realcompactifications
100%
Redlin L. , Watson S.
|
|
tom 152
|
nr 2
151-163
EN
Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).
6
Content available remote Finite Embeddability of Sets and Ultrafilters
100%
Blass A. , Di Nasso M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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tom Vol. 63, no. 3
195--206
EN
A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Čech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.
7
Content available remote Cohen algebras and nowhere dense ultrafilters
88%
Błaszczyk A. , Szymański A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2001
|
tom Vol. 49, no 1
15--25
EN
We study a class of countable extremally disconnected spaces vrithout isolated points. We show that the Cech-Stone compactification of a space from this class admits a semi-open continuous maps onto a Cantor cube if and only if the space itself is modelled by an ultrafilter on omega that is not nowhere dense.
8
Content available Compounding Objects
88%
Šikić Z.
Bulletin of the Section of Logic
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2020
|
tom 49
|
nr 2
EN
We prove a characterization theorem for filters, proper filters and ultrafilters which is a kind of converse of Łoś's theorem. It is more natural than the usual intuition of these terms as large sets of coordinates, which is actually unconvincing in the case of ultrafilters. As a bonus, we get a very simple proof of Łoś's theorem.
9
Content available remote Forcing for First-Order Languages from the Perspective of Rasiowa-Sikorski Lemma
75%
Czelakowski J.
Fundamenta Informaticae
|
2017
|
tom Vol. 156, nr 3/4
255--280
EN
The paper is concerned with the problem of building models for first-order languages from the perspective of the classic paper of Rasiowa and Sikorski [9]. The central idea, developed in this paper, consists in constructing first-order models from individual variables. The key notion of a Rasiowa–Sikorski set of formulas for an arbitrary countable language L is examined. Each Rasiowa–Sikorski set defines a countable model for L. Conversely, every countable model for L is determined by a Rasiowa-Sikorski set. The focus is on constructing Rasiowa-Sikorski sets by applying forcing techniques restricted to Boolean algebras arising from the subsets of the set of atomic formulas of L.
10
Content available remote A new approach to bounded linear operators on C(ω*)
63%
Grzech M.
Czasopismo Techniczne. Nauki Podstawowe
|
2014
|
tom R. 111, z. 3-NP
5--8
EN
We discuss recent results on the connection between properties of a given bounded linear operator of C(ω*) and topological properties of some subset of ω* which the operator determines. A family of closed subsets of ω*, which codes some properties of the operator is defined. An example of application of the method is presented.
PL
Artykuł przedstawia metodę badania własności ograniczonego operatora liniowego na C(ω*) poprzez badanie własności pewnej rodziny domkniętych pozbiorów ω* wyznaczonej przez ten operator. Przedstawiony został przykład zastosowania tej metody w przypadku projekcji.
11
Content available remote Alexander subbase theorem for filters
63%
Labuda I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom Vol. 54, no 2
147-152
EN
The theorem in the title is proven. Applications to product theorems are given.
12
Content available remote Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem
51%
Keremedis K.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom Vol. 53, no 4
349-359
EN
Let X be an infinite set, and P(X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of P(X) can be extended to an ultrafilter. UF(X): P(X) has a free ultrafilter. We will show in ZF (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product 2R, where 2 is the discrete space {0, 1}, is compact. (iii) The Tychonoff product [0, 1] R is compact. (iv) In a Boolean algebra of size ≤ |R| every filter can be extended to an ultrafilter. We will also show that in ZF, UF(R) does not imply BPI(R). Hence, BPI(R) is strictly stronger than UF(R). We do not know if UF(ω) implies BPI(ω) in ZF. Furthermore, we will prove that the axiom of choice for sets of subsets of R does not imply BPI(R) and, in addition, the axiom of choice for well orderable sets of non-empty sets does not imply BPI(ω).
13
Crowded and selective ultrafilters under the covering property axiom
44%
Ciesielski K. , Pawlikowski J.
Journal of Applied Analysis
|
2003
|
tom Vol. 9, nr 1
19-55
EN
In the paper we formulate an axiom CPAgame prism, which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to ω 1 , while c = ω 2 : the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPAgame prism, implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under CPAgame prism every selective ultrafilter is ω 1-generated. The paper finishes with the proof that CPAgame prism holds in the iterated perfect set model.
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