An analogue for semigroups of a group problem of P. Erdős and B. H. Neumann (Q2716960)
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scientific article; zbMATH DE number 1599686
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | An analogue for semigroups of a group problem of P. Erdős and B. H. Neumann |
scientific article; zbMATH DE number 1599686 |
Statements
2 April 2002
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PE-groups
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PE-rings
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centralizers of finite index
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\(0\)-disjoint unions of symmetric groups
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cancellative PE-semigroups
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sets of pairwise non-commuting elements
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An analogue for semigroups of a group problem of P. Erdős and B. H. Neumann (English)
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A semigroup or ring is said to have the PE-property if every set of pairwise non-commuting elements is finite. It is known that every PE-group or PE-ring has a finite bound on the size of the sets of pairwise non-commuting elements (in both cases these algebras are characterized through having centralizers of finite index). This result does not hold for arbitrary semigroups: a counterexample is furnished by a \(0\)-disjoint union of symmetric groups. The main result of this paper is that any cancellative PE-semigroup is group embeddable and has a bound on the cardinality of its sets of pairwise non-commuting elements.
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