Semigroups satisfying a strong Følner condition

Argabright, L. N.; Wilde, Carroll O.

doi:10.1090/s0002-9939-1967-0210797-2

Cited by 15 publications

(7 citation statements)

References 11 publications

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“…We define functions μ and ν on C (S) and C (T ) respectively by putting μ( 1 and Y 2 are disjoint subsets of T . Furthermore, we claim that μ and ν are left invariant.…”

Section: Proof By Lemma 31 S × T Satisfies Sfc and D( A × B) D( A)mentioning

confidence: 99%

“…A left invariant mean on S is a mean μ such that for all s ∈ S and all g ∈ l ∞ (S), μ(s · g) = μ(g), where s · g = g • λ s and λ s : S → S is defined by λ s (t) = st. For any set X , let P f (X) be the set of finite nonempty subsets of X . In [1] Argabright and Wilde showed that a left cancellative semigroup S is left amenable if and only if it satisfies the Strong Følner Condition:…”

Section: Introductionmentioning

confidence: 99%

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Density and invariant means in left amenable semigroups

Hindman

Strauss

2009

Topology and its Applications

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A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any > 0, there is a finite nonempty subset F of S such that for each s ∈ H, |sF F | < · |F |. This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net F α α∈D of finite nonempty subsets of S such that for each s ∈ S, (|sF α F α |)/|F α | converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM 0 (S) which is the set of left invariant means which are weak * limits in l ∞ (S) * of left Følner nets. We show that the set of all left invariant means is the weak * closure of the convex hull of LIM 0 (S).(If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on (N, +) is actually equal to LIM 0 (N). We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on β S, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of β S.

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“…We define functions μ and ν on C (S) and C (T ) respectively by putting μ( 1 and Y 2 are disjoint subsets of T . Furthermore, we claim that μ and ν are left invariant.…”

Section: Proof By Lemma 31 S × T Satisfies Sfc and D( A × B) D( A)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Density and invariant means in left amenable semigroups

Hindman

Strauss

2009

Topology and its Applications

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“…L is called an invariant mean for G. For countable left cancellative semigroups, the existence of Følner sequences and amenability are equivalent [1]. See [32] for a comprehensive overview of the notion of amenability.…”

Section: Preliminariesmentioning

confidence: 99%

“…It will be convenient to also give Cantor spaces a metric space structure. Giving each X n the discrete metric δ n , the space X can be made into a compact metric space via the product metric δ given by (1) δ…”

Section: Cantor Spacesmentioning

confidence: 99%

“…The partition regularity of IP sets is a famous theorem of Hindman [28], and the partition regularity of piecewise syndeticity is a classical fact, often rediscovered (see, e.g., [29,Lemma 2.4] and [22,Theorem 1.24]). 1 The partition regularity of positive upper density is an easy exercise for the reader, as is checking that thickness and syndeticity fail to be partition regular.…”

Section: Introductionmentioning

confidence: 99%

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Multiplicatively large sets and ergodic Ramsey theory

Bergelson

2005

Isr. J. Math.

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We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi-) groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemerédi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. • We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. • We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in Z, Z d , and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. • We show that, for any countable abelian group G, any Følner sequence Φ in G, and any c ∈ (0, 1), there exists a discordant set A ⊆ G with d Φ (A) = c. Here d Φ denotes density along Φ. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.

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A counterexample to Sorenson’s conjecture: The finitely generated case

Yuji

2002

Semigroup Forum

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Semigroups satisfying a strong Følner condition

Cited by 15 publications

References 11 publications

Density and invariant means in left amenable semigroups

Density and invariant means in left amenable semigroups

Multiplicatively large sets and ergodic Ramsey theory

A counterexample to Sorenson’s conjecture: The finitely generated case

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