Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

Baglini, Lorenzo Luperi

doi:10.1002/malq.201800089

Cited by 5 publications

(4 citation statements)

References 24 publications

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“…However, there is some regularity in this respect, given by the next two results. The first of them is a direct corollary of [13, Proposition 4.6]. Proposition For all

F, G \in β N

the set

{x y : x \in μ (F), y \in μ (G)}

is a union of monads.…”

Section: More About Monadsmentioning

confidence: 99%

“…In [11] and [13] several theorems were proved that enable us to translate formulas from

V (^{*} double-struckN)

to equivalent formulas in

V (N)

and vice versa. The basic such theorem was called The Bridge Theorem there, so we will address all such results as bridge theorems.…”

Section: More About Monadsmentioning

confidence: 99%

“…Such pairs

(x, y)

are called tensor pairs . They were introduced implicitly in [16] and further investigated in [3, 11, 13]. They will also become an important tool for the research of divisibility in a follow‐up paper.…”

Section: More About Monadsmentioning

confidence: 99%

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More about divisibility in βN

Šobot

2021

Mathematical Logic Qtrly

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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 double-struckN are answered, providing a few more equivalent conditions for divisibility in βN. Results on uncountable chains in (βN,true∣∼) are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type (R,<) in (βN,true∣∼). Finally, we consider ultrafilters without divisors in double-struckN and among them find the greatest class.

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“…However, there is some regularity in this respect, given by the next two results. The first of them is a direct corollary of [13, Proposition 4.6]. Proposition For all

F, G \in β N

the set

{x y : x \in μ (F), y \in μ (G)}

is a union of monads.…”

Section: More About Monadsmentioning

confidence: 99%

“…In [11] and [13] several theorems were proved that enable us to translate formulas from

V (^{*} double-struckN)

to equivalent formulas in

V (N)

and vice versa. The basic such theorem was called The Bridge Theorem there, so we will address all such results as bridge theorems.…”

Section: More About Monadsmentioning

confidence: 99%

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More about divisibility in βN

Šobot

2021

Mathematical Logic Qtrly

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“…1 In this case, we call (a, b) a tensor pair. Tensor pairs in * N have been characterised by Puritz in [Pur71]; we recall here the extension of Puritz' characterisation to * Z, and we refer to [DN15, Section 11.5] or [LB19] for a proof of this fact. The iterated hyper-extensions framework of nonstandard analysis allows for an even simpler characterisation of tensor products and related notions: if a, b ∈ * Z are such that a |= u and b |= v, then (a, * b) |= u ⊗ v. As a trivial consequence, in the same hypotheses we have that a + * b |= u ⊕ v and a • * b |= u v. A detailed study of many properties and characterisations of tensor k-uples in this iterated nonstandard context can be found in [LB19].…”

Section: Preliminariesmentioning

confidence: 99%

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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Partition regularity of polynomial systems near zero

Baglini

2021

Semigroup Forum

Self Cite

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Recently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups $$(S,+)$$ ( S , + ) of $$(\mathbb {R}^{+},+)$$ ( R + , + ) . We extend their results to a much larger class of systems whilst weakening the requests on S, using solely basic results about ultrafilters.

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Nonstandard characterisations of tensor products and monads in the theory of ultrafilters

Cited by 5 publications

References 24 publications

More about divisibility in βN

More about divisibility in βN

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

Partition regularity of polynomial systems near zero

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