Multiplicative finite embeddability vs divisibility of ultrafilters

Šobot, Boris

doi:10.1007/s00153-021-00799-y

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…In the case of βN, the following is [ Šob22,Lemma 4.3]. Its version for βZ is proven in the same way.…”

Section: Preliminariesmentioning

confidence: 82%

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

Nasso¹,

Baglini²,

Mennuni³

et al. 2023

Preprint

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We introduce self-divisible ultrafilters, which we prove to be precisely those w such that the weak congruence relation ≡w introduced by Šobot is an equivalence relation on βZ. We provide several examples and additional characterisations; notably we show that w is self-divisible if and only if ≡w coincides with the strong congruence relation ≡ s w , if and only if the quotient (βZ, ⊕)/≡ s w is a profinite group. We also construct an ultrafilter w such that ≡w fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion Ẑ of the integers.In [ Šob21], Šobot investigated generalisations of the congruence relation a ≡ n b from Z to its Stone-Čech compactification βZ, equipped with the usual extensions ⊕, of the sum and product of integers. For each w ∈ βN, he introduced a congruence relation ≡ w and a strong congruence relation ≡ s w . In this paper we investigate these notions and prove that, for some w, the former fails to be an equivalence relation, thereby answering [ Šob21, Question 7.1] in the negative. In fact, we fully characterise those w for which this happens, and compute the relative quotient when it does not.Almost by definition, u ≡ s w v holds if and only if, whenever (d, a, b) is an ordered triple of nonstandard integers which generate w ⊗ u ⊗ v, we have d | a − b. It was proven in [ Šob21] that ≡ s w is always an equivalence relation, and in fact a congruence with respect to ⊕ and but, perhaps counterintuitively for a notion of congruence, there are some ultrafilters w for which w ≡ s w 0. On the other hand, the relation ≡ w does always satisfy w ≡ w 0 but, as we said above, it may fail to be an equivalence relation. So, in a sense, these two relations have complementary drawbacks, and it is natural to ask for which ultrafilters w these drawbacks disappear. Our main result says that ≡ w is well-behaved if and only if ≡ s w is, if and only if the two relations collapse onto each other. This is moreover equivalent to the quotient (βZ, ⊕)/≡ s w being a profinite group, which in fact can be explicitly computed. More precisely, if we denote by P the set of prime natural numbers and by Z p the additive group of p-adic integers, our main results can be summarised as follows.Main Theorem (Theorems 3.10 and 6.8). For every w ∈ βN the following are equivalent.(1) We have w ≡ s w 0.(2) The relation ≡ w is an equivalence relation.(3) The relations ≡ w and ≡ s w coincide.

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“…In the case of βN, the following is [ Šob22,Lemma 4.3]. Its version for βZ is proven in the same way.…”

Section: Preliminariesmentioning

confidence: 82%

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

Nasso¹,

Baglini²,

Mennuni³

et al. 2023

Preprint

View full text Add to dashboard Cite

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“…More results about the "lower" part of the | -hierarchy can be found in [10]. What about the "upper" part?…”

Section: Introductionmentioning

confidence: 99%

More number theory in $βN$

Šobot¹

2019

Preprint

Self Cite

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We continue the research of an extension | of the divisibility relation to the Stone-Čech compactification βN . First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in βN and nonstandard extensions of N are answered, providing a few more equivalent conditions for divisibility in βN . Results on uncountable chains in (βN, | ) are proved and used in a construction of a well-ordered chain of maximal cardinality. Finally, we consider ultrafilters without divisors in N and among them find the maximal class.

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Self-Divisible Ultrafilters and Congruences In

Nasso¹,

Baglini²,

Mennuni³

et al. 2023

J. symb. log.

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We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv _w$ introduced by Šobot is an equivalence relation on $\beta {\mathbb Z}$ . We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv _w$ coincides with the strong congruence relation $\mathrel {\equiv ^{\mathrm {s}}_{w}}$ , if and only if the quotient $(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv _w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat {{\mathbb Z}}$ of the integers.

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Multiplicative finite embeddability vs divisibility of ultrafilters

Cited by 3 publications

References 8 publications

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

More number theory in $βN$

Self-Divisible Ultrafilters and Congruences In

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