On locally finite semigroups

Brown, Thomas K.

doi:10.1007/bf01085231

Cited by 22 publications

(24 citation statements)

References 4 publications

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“…X r−1 be a finite partition. Then X i is piecewise syndetic for some i < r. Theorem 1.2 was originally proved by Brown [Bro68] (see also [Bro69,Bro71]) in the context of locally finite semigroups. An ergodic-theoretic proof of Theorem 1.2 as well as Theorem 1.4 can be found in Furstenberg [Fur81, Theorem 1.23, Theorem 1.24].…”

Section: Introductionmentioning

confidence: 91%

Brown’s lemma in second-order arithmetic

Frittaion

2017

Fund. Math.

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Abstract. Brown's lemma states that in every finite coloring of the natural numbers there is a homogeneous piecewise syndetic set. We show that Brown's lemma is equivalent to IΣ 0 2 over RCA * 0 . We show in contrast that (the infinite) van der Waerden's theorem is equivalent to BΣ 0 2 over RCA * 0 . We finally consider the finite version of Brown's lemma and show that it is provable in RCA 0 but not in RCA * 0 .

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Section: Introductionmentioning

confidence: 91%

Brown’s lemma in second-order arithmetic

Frittaion

2017

Fund. Math.

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“…The first two parts are easy exercises, while the third follows from a wellknown result of Brown [13,14].…”

Section: Some Easy Propertiesmentioning

confidence: 99%

Presentations of Inverse Semigroups, Their Kernels and Extensions

Carvalho¹,

Gray²,

Ruškuc³

2011

J. Aust. Math. Soc.

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Let S be an inverse semigroup and let π : S → T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S , and vice versa. We then investigate the relationship between the properties of S , K and T , focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FP n . Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.2010 Mathematics subject classification: primary 20M18; secondary 20M05.

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“…The families Sm G and Sp G of all small and sparse subsets of G are ideals in the Boolean algebra P G (see [2] and [7] respectively). Recall that a subset I ⊆ P G is an ideal if I is closed under taking subsets and finite unions (we do not suppose that G / ∈ I).…”

Section: Introductionmentioning

confidence: 99%

The combinatorial derivation and its inverse mapping

Протасов

2013

Open Mathematics

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Let G be a group and P G be the Boolean algebra of all subsets of G. A mapping ∆ :A ∩ A is infinite} is called the combinatorial derivation. The mapping ∆ can be considered as an analogue of the topological derivation : P X → P X , A → A , where X is a topological space and A is the set of all limit points of A. We study the behaviour of subsets of G under action of ∆ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in P G such that ∆(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G . MSC:20A05

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On locally finite semigroups

Cited by 22 publications

References 4 publications

Brown’s lemma in second-order arithmetic

Brown’s lemma in second-order arithmetic

Presentations of Inverse Semigroups, Their Kernels and Extensions

The combinatorial derivation and its inverse mapping

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