Semigroups of transformations restricted by an equivalence

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)...

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Bibliographic Details
Published in:Central European journal of mathematics 2010-12, Vol.8 (6), p.1120-1131
Main Authors: Mendes-Gonçalves, Suzana, Sullivan, Robert P.
Format: Article
Language:English
Subjects:
20M20
Algebra
Composition
Equivalence
Geometry
Green’s relations
Ideals
Lie Groups
Linear transformations
Mathematics
Mathematics and Statistics
Number Theory
Probability Theory and Stochastic Processes
Research Article
Semigroups
Topological Groups
Transformation semigroup
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Summary: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 .
ISSN:1895-1074
1644-3616
2391-5455
DOI:10.2478/s11533-010-0066-8