Semigroups of transformations restricted by an equivalence

Suzana Mendes-Gonçalves; Robert Sullivan

Open Mathematics (2010)

  • Volume: 8, Issue: 6, page 1120-1131
  • ISSN: 2391-5455

Abstract

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Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.

How to cite

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Suzana Mendes-Gonçalves, and Robert Sullivan. "Semigroups of transformations restricted by an equivalence." Open Mathematics 8.6 (2010): 1120-1131. <http://eudml.org/doc/269067>.

@article{SuzanaMendes2010,
abstract = {Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.},
author = {Suzana Mendes-Gonçalves, Robert Sullivan},
journal = {Open Mathematics},
keywords = {Transformation semigroup; Equivalence; Green’s relations; Ideals; transformation semigroups; equivalences; Green relations},
language = {eng},
number = {6},
pages = {1120-1131},
title = {Semigroups of transformations restricted by an equivalence},
url = {http://eudml.org/doc/269067},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Suzana Mendes-Gonçalves
AU - Robert Sullivan
TI - Semigroups of transformations restricted by an equivalence
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 1120
EP - 1131
AB - Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.
LA - eng
KW - Transformation semigroup; Equivalence; Green’s relations; Ideals; transformation semigroups; equivalences; Green relations
UR - http://eudml.org/doc/269067
ER -

References

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  1. [1] Clifford A.H., Preston G.B., The Algebraic Theory of Semigroups, Vol. 1 and 2, Math. Surveys Monogr., 7, American Mathematical Society, Providence, 1961 and 1967 Zbl0111.03403
  2. [2] Howie J.M., Fundamentals of Semigroup Theory, London Math. Soc. Monogr. Ser., 12, Clarendon Press, Oxford, 1995 Zbl0835.20077
  3. [3] Pei H., Regularity and Green’s relations for semigroups of transformations that preserve an equivalence, Comm. Algebra, 2005, 33(1), 109–118 http://dx.doi.org/10.1081/AGB-200040921 Zbl1072.20082
  4. [4] Pei H., Deng W., A note on Green’s relations in the semigroups T(X, ρ), Semigroup Forum, 2009, 79(1), 210–213 http://dx.doi.org/10.1007/s00233-009-9151-3 Zbl1172.20046
  5. [5] Sullivan R.P., Semigroups of linear transformations with restricted range, Bull. Austral. Math. Soc., 2008, 77(3), 441–453 http://dx.doi.org/10.1017/S0004972708000385 Zbl1149.20050
  6. [6] Sullivan R.P., Semigroups of linear transformations with restricted kernel (submitted) 

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