Large rectangular semigroups in Stone-Cech compactifications

Hindman, Neil; Strauss, Dona; Zelenyuk, Yevhen

doi:10.1090/s0002-9947-03-03276-8

Cited by 8 publications

(4 citation statements)

References 11 publications

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“…Equivalently, e < L f if and only if Se is a proper subset of Sf . Using a result from [6] we show here that one can get chains as long as possible below any nonminimal idempotent in βN, provided that one weakens the strictly decreasing requirement at limit ordinals to the requirement that if σ < τ , then p τ < L p σ . We will use the following extension of Lemma 2.3.…”

Section: Countable Left Cancellative Semigroupsmentioning

confidence: 90%

“…Besides the ordering ≤ of idempotents in a semigroup, there are transitive and reflexive relations ≤ L and ≤ R defined by e ≤ L f if ef = e, and e ≤ R f if f e = e. We write e < L f when e ≤ L f and it is not the case that f ≤ L e. Similarly we write e < R f when e ≤ R f and it is not the case that f ≤ R e. Of course e ≤ f if and only if both e ≤ L f and e ≤ R f . In [6] it was shown that given any ordinal λ with |λ| ≤ c, there exist chains q σ σ<λ of idempotents in βN such that q σ < L q τ whenever τ < σ < λ and q σ+1 < q σ for all σ with σ + 1 < λ. In Section 5 we extend this result by showing that for each nonminimal idempotent q in βN, there is such a chain with q 0 = q.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of the argument may be taken verbatim from the proof that (5) implies (6) in [8,Theorem 11.2].…”

Section: Introductionmentioning

confidence: 99%

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Longer chains of idempotents in βG

2013

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Given idempotents e and f in a semigroup, e ≤ f if and only if e = f e = ef . We show that if G is a countable discrete group, p is a right cancelable element of G * = βG \ G, and λ is a countable ordinal, then there is a strictly decreasing chain qσ σ<λ of idempotents in Cp, the smallest compact subsemigroup of G * with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S * is the largest element of such a strictly decreasing chain of idempotents. (It had been an open question whether there was a strictly decreasing chain qσ σ<ω+1 in N * .) As other corollaries we show that if S is an infinite right cancellative and weakly left cancellative discrete semigroup, then βS contains a decreasing chain of idempotents of reverse order type λ for every countable ordinal λ and that if S is an infinite cancellative semigroup then the set U (S) of uniform ultrafilters contains such decreasing chains.

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Section: Countable Left Cancellative Semigroupsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Longer chains of idempotents in βG

2013

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“…Proof. The first assertion is an immediate consequence of [9,Corollary 3.15]. Let H = ∞ n=1 2 n N. By [8, Lemma 6.8], H is a compact subsemigroup of (βN, +) which contains all of the idempotents.…”

Section: Theorem 22mentioning

confidence: 97%

Some new multi-cell Ramsey theoretic results

Bergelson

Hindman

2021

Proc. Amer. Math. Soc. Ser. B

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We extend an old Ramsey Theoretic result which guarantees sums of terms from all partition regular linear systems in one cell of a partition of the set N \mathbb {N} of positive integers. We were motivated by a quite recent result which guarantees a sequence in one set with all of its sums two or more at a time in the complement of that set. A simple instance of our new results is the following. Let P f ( N ) \mathcal {P}_{f}(\mathbb {N}) be the set of finite nonempty subsets of N \mathbb {N} . Given any finite partition R {\mathcal R} of N \mathbb {N} , there exist B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} in R {\mathcal R} and sequences ⟨ x 1 , n ⟩ n = 1 ∞ \langle x_{1,n}\rangle _{n=1}^\infty and ⟨ x 2 , n ⟩ n = 1 ∞ \langle x_{2,n}\rangle _{n=1}^\infty in N \mathbb {N} such that (1) for each F ∈ P f ( N ) F\in \mathcal {P}_{f}(\mathbb {N}) , ∑ t ∈ F x 1 , t ∈ B 1 \sum _{t\in F}x_{1,t}\in B_1 and ∑ t ∈ F x 2 , t ∈ B 2 \sum _{t\in F}x_{2,t}\in B_2 and (2) whenever F , G ∈ P f ( N ) F,G\in \mathcal {P}_{f}(\mathbb {N}) and max F > min G \max F > \min G , one has ∑ t ∈ F x 1 , t + ∑ t ∈ G x 2 , t ∈ A 1 , 2 \sum _{t\in F}x_{1,t}+\sum _{t\in G}x_{2,t}\in A_{1,2} and ∑ t ∈ F x 2 , t + ∑ t ∈ G x 1 , t ∈ A 2 , 1 \sum _{t\in F}x_{2,t}+\sum _{t\in G}x_{1,t}\in A_{2,1} . The partition R {\mathcal R} can be refined so that the cells B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} must be pairwise disjoint.

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Ultrafilter semigroups generated by direct sums

Shuungula

Zelenyuk

2010

Semigroup Forum

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Large rectangular semigroups in Stone-Cech compactifications

Cited by 8 publications

References 11 publications

Longer chains of idempotents in βG

Longer chains of idempotents in βG

Some new multi-cell Ramsey theoretic results

Ultrafilter semigroups generated by direct sums

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