Sur les suites infinies de fonctions définies dans les ensembles quelconques

Sierpiński, Wacław

doi:10.4064/fm-24-1-209-212

Cited by 53 publications

(34 citation statements)

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“…Surprisingly, the answer is 'no'. That conclusion can be deduced from Proposition 1.1, first proved by Sierpiński in 1935 [10]. A simpler proof was immediately furnished however by Banach [1].…”

Section: Introductionmentioning

confidence: 83%

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Higgins¹,

Howie²,

Mitchell³

et al. 2003

Proceedings of the Edinburgh Mathematical Society

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The relative rank rank(S : A) of a subset A of a semigroup S is the minimum cardinality of a set B such that A ∪ B = S. It follows from a result of Sierpiński that, if X is infinite, the relative rank of a subset of the full transformation semigroup T X is either uncountable or at most 2. A similar result holds for the semigroup B X of binary relations on X.A subset S of T N is dominated (by U ) if there exists a countable subset U of T N with the property that for each σ in S there exists µ in U such that iσ iµ for all i in N. It is shown that every dominated subset of T N is of uncountable relative rank. As a consequence, the monoid of all contractions in T N (mappings α with the property that |iα − jα| |i − j| for all i and j) is of uncountable relative rank.It is shown (among other results) that rank(B X : T X ) = 1 and that rank(B X : I X ) = 1 (where I X is the symmetric inverse semigroup on X). By contrast, if S X is the symmetric group, rank(B X : S X ) = 2.

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Section: Introductionmentioning

confidence: 83%

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Higgins¹,

Howie²,

Mitchell³

et al. 2003

Proceedings of the Edinburgh Mathematical Society

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show abstract

“…An alternative proof can be obtained using Lemma 2.4. It follows from [24], and [12, Proposition 4.2] that for all sequences (f n ) n∈N of elements from S there exist f, g ∈ S such that every f n is a product of f and g wih length bounded by a linear function. Hence S is strongly distorted and so, by Lemma 2.4, scf(S) > ℵ 0 .…”

Section: Positive Examplesmentioning

confidence: 99%

The Bergman property for semigroups

Maltcev

Mitchell

Ruškuc

2009

Journal of the London Mathematical Society

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In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa.Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.

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“…In the same paper it was shown that the relative rank of the set of all idempotent maps on X in T X is, also, two. Sierpiński [15] showed that any countable set of maps from X to X is contained in a 2-generated subsemigroup of T X . An alternative proof of this was given by Banach [1]; see also [8].…”

Section: Introductionmentioning

confidence: 99%

Generating the Full Transformation Semigroup Using Order Preserving Mappings

Higgins¹,

Mitchell

Ruškuc³

2003

Glasgow Math. J.

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Abstract. For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O X on X modulo the full transformation semigroup T X . In other words, we ask what is the smallest cardinality of a set A of mappings such that O X ∪ A = T X . When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = ‫ޒ‬ (the set of real numbers) it is uncountable.2000 Mathematics Subject Classification. 20M20, 06A05.

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Sur les suites infinies de fonctions définies dans les ensembles quelconques

Cited by 53 publications

References 0 publications

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

Countable Versus Uncountable Ranks in Infinite Semigroups of Transformations and Relations

The Bergman property for semigroups

Generating the Full Transformation Semigroup Using Order Preserving Mappings

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