# 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

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topSuzana 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

top- [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] Howie J.M., Fundamentals of Semigroup Theory, London Math. Soc. Monogr. Ser., 12, Clarendon Press, Oxford, 1995 Zbl0835.20077
- [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] 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] 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] Sullivan R.P., Semigroups of linear transformations with restricted kernel (submitted)

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