The relationships among many notions of largeness for subsets of a semigroup

Hindman, Neil; Jones, Lakeshia Legette; Strauss, Dona

doi:10.1007/s00233-018-9944-3

Cited by 3 publications

(1 citation statement)

References 8 publications

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“…Several results in this section use the hypothesis that there is a uniform, finite bound on the size of the sets Fix(a). The left-right switch of [6,Theorem 4.11] shows that this assumption is strictly weaker than the assertion that there is a finite bound on the size of right solution sets. Proof.…”

Section: Topology In K(βs)mentioning

confidence: 99%

Factoring a minimal ultrafilter into a thick part and a syndetic part

Brian¹,

Hindman²

2018

Preprint

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Let S be an infinite discrete semigroup. The operation on S extends uniquely to the Stone-Čech compactification βS making βS a compact right topological semigroup with S contained in its topological center. As such, βS has a smallest two sided ideal, K(βS). An ultrafilter p on S is minimal if and only if p ∈ K(βS).We show that any minimal ultrafilter p factors into a thick part and a syndetic part. That is, there exist filters F and G such that F consists only of thick sets, G consists only of syndetic sets, and p is the unique ultrafilter containing F ∪ G.Letting L = F and C = G, the sets of ultrafilters containing F and G respectively, we have that L is a minimal left ideal of βS, C meets every minimal left ideal of βS in exactly one point, and L ∩ C = {p}. We show further that K(βS) can be partitioned into relatively closed sets, each of which meets each minimal left ideal in exactly one point.With some weak cancellation assumptions on S, one has also that for each minimal ultrafilter p, S * \ {p} is not normal. In particular, if p is a member of either of the disjoint sets K(βN, +) or K(βN, •), then N * \ {p} is not normal.

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Section: Topology In K(βs)mentioning

confidence: 99%

Factoring a minimal ultrafilter into a thick part and a syndetic part

Brian¹,

Hindman²

2018

Preprint

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Notions of size in a semigroup: an update from a historical perspective

Hindman

2019

Semigroup Forum

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Previous papers have investigated relationships among several notions of largeness in a semigroup, some of which have their origins in topological dynamics, others with pure combinatorial roots, and still others based on the algebraic structure of the Stone-Cech compactification of a discrete semigroup. Here we consider 52 distinct notions of largeness, giving to the extent possible a description of the origins and why the notions are of interest. We establish implications that must hold among these notions. In the event the semigroup is commutative, these reduce to 24 distinct notions. We give examples in (N, +) showing that the notions satisfy only the implications which we have established for commutative semigroups.

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Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

Glasscock

2024

Advances in Mathematics

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The relationships among many notions of largeness for subsets of a semigroup

Cited by 3 publications

References 8 publications

Factoring a minimal ultrafilter into a thick part and a syndetic part

Factoring a minimal ultrafilter into a thick part and a syndetic part

Notions of size in a semigroup: an update from a historical perspective

Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

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