A propos de la propriété de Bergman

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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“…, t N S . The following lemma was suggested to us by Y. Cornulier and a similar result appears in Khelif [17,Theorem 6].…”

Section: Cofinality and Strong Cofinalitymentioning

confidence: 91%

“…Khelif [17] provided an example of a group G where the least cardinal of a cofinal chain of subgroups for G is ℵ 0 and that satisfies the group Bergman property. Using the same reasoning as in Example 2.6 we deduce that Khelif's group satisfies (iii).…”

Section: Cofinality and Strong Cofinalitymentioning

confidence: 99%

The Bergman property for semigroups

Maltcev

Mitchell

Ruškuc

2009

Journal of the London Mathematical Society

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“…Note that an uncountable group with cofinality = ω is not necessarily Cayley bounded: the free product of two uncountable groups of cofinality = ω, or the direct product of an uncountable group of cofinality = ω with Z, are obvious counterexamples. On the other hand, a Cayley bounded group with cofinality ω is announced in [Khe05].…”

Section: Strongly Bounded Groupsmentioning

confidence: 99%

Strongly Bounded Groups and Infinite Powers of Finite Groups

Cornulier¹

2006

Communications in Algebra

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Abstract. We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently introduced by Bergman.Our main result is that G I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω 1 -existentially closed groups are strongly bounded.

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“…Khelif [16,Définition 5] introduces a condition having an interesting similarity to (32). Given an algebra-type T and a natural number n, let W (n, T ) denote the set of all n-ary words in the operations of T. Note that if S is an algebra of type T, then an element of W (n, T ) ω induces an n-ary operation on S ω .…”

Section: Proof (I)mentioning

confidence: 99%

“…Let us fix i 0 for the remainder of the proof, and establish (16). We first note that (15) entails the weaker statement gotten by ignoring cases before the i 0 th, and replacing all the ideals I i (i ≥ i 0 ) with the larger ideal I i0 :…”

mentioning

confidence: 99%

Two statements about infinite products that are not quite true

Bergman¹

2006

Contemporary Mathematics

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infinite product module, inverse limit module, dichotomy of finite or uncountable generation, homomorphism to infinite direct sum, left perfect ring; infinite symmetric group, finite generation of power or ultrapower of an algebra over the diagonal subalgebra.URLs of this preprint: http://math.berkeley.edu/∼ gbergman/papers/P vs cP.{tex,dvi,ps, pdf} (i.e., four versions, .../.tex, .../.dvi, .../.ps and .../.pdf).Comments, corrections, and related references are welcomed, as always!

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A propos de la propriété de Bergman

Cited by 16 publications

References 6 publications

The Bergman property for semigroups

The Bergman property for semigroups

Strongly Bounded Groups and Infinite Powers of Finite Groups

Two statements about infinite products that are not quite true

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