Chains of idempotents in $\beta\bold N$

Hindman, Neil; Strauss, Dona

doi:10.1090/s0002-9939-1995-1301502-4

Cited by 4 publications

(5 citation statements)

References 16 publications

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“…We are now able to establish a considerably stronger statement. (The necessity in the following corollary was also established in [5], but we include the short proof for completeness.) Sufficiency.…”

Section: Theorem 314 Let T Be a Compact Right Topological Semigroupmentioning

confidence: 82%

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Large rectangular semigroups in Stone-Cech compactifications

Hindman

Strauss

Zelenyuk

2003

Trans. Amer. Math. Soc.

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Abstract. We show that large rectangular semigroups can be found in certain Stone-Čech compactifications. In particular, there are copies of the 2 c × 2 c rectangular semigroup in the smallest ideal of (βN, +), and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of (βN, +) if and only if it is a subsemigroup of the 2 c × 2 c rectangular semigroup. In fact, we show that for any ordinal λ with cardinality at most c, βN contains a semigroup of idempotents whose rectangular components are all copies of the 2 c × 2 c rectangular semigroup and form a decreasing chain indexed by λ + 1, with the minimum component contained in the smallest ideal of βN.As a fortuitous corollary we obtain the fact that there are ≤ L -chains of idempotents of length c in βN. We show also that there are copies of the direct product of the 2 c × 2 c rectangular semigroup with the free group on 2 c generators contained in the smallest ideal of βN.

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Section: Theorem 314 Let T Be a Compact Right Topological Semigroupmentioning

confidence: 82%

“…It was shown in [5,Corollary 3.4] that there is a ≤ L -chain u σ σ<ω1 of distinct idempotents in βN with the property that for each σ < ω 1 , u σ+1 ≤ u σ . We are now able to establish a considerably stronger statement.…”

Section: Theorem 314 Let T Be a Compact Right Topological Semigroupmentioning

confidence: 99%

Large rectangular semigroups in Stone-Cech compactifications

Hindman

Strauss

Zelenyuk

2003

Trans. Amer. Math. Soc.

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“…In solving this problem, results of [13] may be useful. The general Problem 5.6 is also interesting in special cases, for example, where X is a (semi)group, the action is assumed to satisfy additional conditions (e.g., be transitive, faithful, or free), or the orbit space has certain properties (e.g., is countable).…”

Section: G-topologizability Of G-actsmentioning

confidence: 99%

Topologizations of a set endowed with an action of a monoid

Банах

Протасов

Sipacheva

2014

Topology and its Applications

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Abstract. Given a set X and a family G of self-maps of X, we study the problem of the existence of a nondiscrete Hausdorff topology on X with respect to which all functions f ∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζ G on X, that is, the topology generated by the subbase consisting of the sets {x ∈ X : f (x) = g(x)} and {x ∈ X : f (x) = c}, where f, g ∈ G and c ∈ X. We prove that, for a countable monoid G ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.

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“…This contrasts with the situation for the Stone-Čech compactification βN of (N, +), where it is not yet known whether increasing infinite sequences of idempotents exist. Conclusions about chains of idempotents in that context are hard to obtain (see [14]). This is probably due to the non-commutativity of the Stone-Čech compactification.…”

Section: The Semigroups S Fmentioning

confidence: 99%

The weakly almost periodic compactification of a direct sum of finite groups

Bordbar

Pym

1998

Math. Proc. Camb. Phil. Soc.

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The success of the methods of [24] and [4] in investigating the structure of the weakly almost periodic compactification wℕ of the semigroup (ℕ, +) of positive integers has prompted us to see how successful they would be in another context. We consider wG, where G=[oplus ]i∈ωGi is the direct sum of a sequence of finite groups with its discrete topology. We discover a large class of weakly almost periodic functions on G, and we use them to prove the existence of a large number of long chains of idempotents in wG. However, the closure of any singly generated subsemigroup of wG contains only one idempotent. We also prove that the set of idempotents in wG is not closed. The minimal idempotent of wG can be written as the sum of two others, with the consequence that the minimal ideal of wG is not prime. Pursuing possible parallels with the structure of βℕ, we find subsemigroups SF of wG which are ‘carried’ by closed subsets F of βω. wG contains the free abelian product of the semigroups SF corresponding to families of disjoint subsets F. Usually SF is a very large semigroup, but for some points p∈βω, Sp can be quite small.

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Chains of idempotents in $\beta\bold N$

Abstract: Abstract. We show that any non-minimal idempotent in the semigroup (/?N, +) lies in a sequence of idempotents each smaller than its predecessor and each maximal among all idempotents smaller than its predecessor.

Cited by 4 publications

References 16 publications

Large rectangular semigroups in Stone-Cech compactifications

Large rectangular semigroups in Stone-Cech compactifications

Topologizations of a set endowed with an action of a monoid

The weakly almost periodic compactification of a direct sum of finite groups

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