Infinite injective transformations whose centralizers have simple structure

Janusz Konieczny

Open Mathematics (2011)

  • Volume: 9, Issue: 1, page 23-35
  • ISSN: 2391-5455

Abstract

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For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = β ∈ g/g(X): αβ = βα be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Green’s relations in C(α) coincide, characterize α ∈ Γ(X) such that the 𝒥 -classes of C(α) form a chain, and describe Green’s relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.

How to cite

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Janusz Konieczny. "Infinite injective transformations whose centralizers have simple structure." Open Mathematics 9.1 (2011): 23-35. <http://eudml.org/doc/269142>.

@article{JanuszKonieczny2011,
abstract = {For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = β ∈ g/g(X): αβ = βα be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Green’s relations in C(α) coincide, characterize α ∈ Γ(X) such that the \[ \mathcal \{J\} \] -classes of C(α) form a chain, and describe Green’s relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.},
author = {Janusz Konieczny},
journal = {Open Mathematics},
keywords = {Infinite injective transformations; Centralizers; Green's relations; infinite injective transformations; semigroups of transformations; centralizers; Green relations; partial orders},
language = {eng},
number = {1},
pages = {23-35},
title = {Infinite injective transformations whose centralizers have simple structure},
url = {http://eudml.org/doc/269142},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Janusz Konieczny
TI - Infinite injective transformations whose centralizers have simple structure
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 23
EP - 35
AB - For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X under function composition. For α ∈ Γ(X), let C(α) = β ∈ g/g(X): αβ = βα be the centralizer of α in Γ(X). The aim of this paper is to determine those elements of Γ(X) whose centralizers have simple structure. We find α ∈ (X) such that various Green’s relations in C(α) coincide, characterize α ∈ Γ(X) such that the \[ \mathcal {J} \] -classes of C(α) form a chain, and describe Green’s relations in C(α) for α with so-called finite ray-cycle decomposition. If α is a permutation, we also find the structure of C(α) in terms of direct and wreath products of familiar semigroups.
LA - eng
KW - Infinite injective transformations; Centralizers; Green's relations; infinite injective transformations; semigroups of transformations; centralizers; Green relations; partial orders
UR - http://eudml.org/doc/269142
ER -

References

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