divisibility of ultrafilters Šobot, Boris
Annals of pure and applied logic,
January 2021, 2021-01-00, Letnik:
172, Številka:
1
Journal Article
Recenzirano
We further investigate a divisibility relation on the set βN of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a ...hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.
Let X be a topological space and let
be an ideal of subsets of X. The space X is called connected modulo
if there is no continuous mapping f : X → 0, 1 which is 2-valued modulo
in the sense that ...neither f
−1
(0) nor f
−1
(1) belongs to
but X \ (f
−1
(0) ∪ f
−1
(1)) belongs to
. We prove that a completely regular space X is connected modulo
if and only if the quotient
of the ring C
B
(X) (of all bounded continuous real-valued mappings on X equipped with pointwise addition and multiplication) is indecomposable. Here
is the ideal of C
B
(X) consisting of all f in C
B
(X) such that |f|
−1
(ε, ∞)) belongs to
for any positive ε. We examine examples corresponding to various choices of the ideal
. We conclude with consideration of the ideal
of CB (X) whose importance is highlighted by our characterization theorem.
We show that there exists a Tychonoff weakly Lindelöf space which does not have the property Ufin⁎(O,O). This result answers the following open questions. 2 Does a weakly Lindelöf space have the ...property U1⁎(O,O)? 3 Is there a Tychonoff 1-star-Lindelöf space which does not have the property Ufin⁎(O,O)?
For a completely regular space X, let CB(X) be the normed algebra of all bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm and ...let C0(X) be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed ideal H of CB(X) we study properties of its spectrum sp(H) which may be characterized as the unique locally compact (Hausdorff) space Y such that H and C0(Y) are isometrically isomorphic. We concentrate on compactness properties of sp(H) and find necessary and sufficient (algebraic) conditions on H such that the spectrum sp(H) satisfies (topological) properties such as the Lindelöf property, σ-compactness, countable compactness, pseudocompactness and paracompactness.
In the framework of finitely additive measures defined on the power set of an infinite set X, we consider related operators defined on bounded functions on X invariant under finitely many changes of ...input values. Specifically, we reconsider these operators with an extended use of the concept of filter, which provides novel insights into the problem. Then, we apply the obtained results to the study of the aggregation of infinite sequences.
Cyclic sequences of periodic sums in βN Keyantuo, Valentin; Zelenyuk, Yevhen; Zelenyuk, Yuliya
Topology and its applications,
01/2024, Letnik:
341
Journal Article
Recenzirano
Let m≥2 and n≥2 and define ν:ω→{0,…,m−1} by ν(k)≡k(modm). We show that there is a sequence p0,…,pm−1 in βN such that all sums ∑j=ikpν(j), where i∈{0,…,m−1} and k∈{i,…,mn−1} for each i, are distinct ...and ∑j=imnpν(j)=∑j=imn−mpν(j) for each i. As a consequence we derive a new Ramsey theoretic result: there is a partition {Ai,k:i∈{0,…,m−1} and k∈{i,…,mn−1} for each i} of N such that, whenever for each (i,k), Bi,k is a finite partition of Ai,k, there exist Bi,k∈Bi,k and a sequence (xj)j=0∞ such that for every finite sequence j0<…<js such that jt+1≡jt+1(modm) for each t<s, one has xj0+…+xjs∈Bi0,k0, where i0=ν(j0) and k0 is i0+s if i0+s≤mn−1 and mn−m+ν(i0+s−mn) otherwise.
We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-Čech remainder obtained by contracting the connected components.
In this paper we deal with surjective linear isometries between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily closed or bounded) subsets of the real line (with ...at least two points). As a corollary, it is shown that when the underlying spaces are connected, each surjective linear isometry of these function spaces is a weighted composition operator, a result which generalizes all the previous known results concerning such isometries.
In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect ...to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace.
We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and ...multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite coloring N=C1∪…∪Cr there exists an infinite increasing sequence a<b<c<… such that all elements below are monochromatic:a,b,c,…,a+b+ab,a+c+ac,b+c+bc,…,a+b+c+ab+ac+bc+abc,…. The proofs use tools from algebra in the space of ultrafilters βZ.
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