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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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
Howie J.M., Fundamentals of Semigroup Theory, London Math. Soc. Monogr. Ser., 12, Clarendon Press, Oxford, 1995
Pei H., Regularity and Green’s relations for semigroups of transformations that preserve an equivalence, Comm. Algebra, 2005, 33(1), 109–118
Pei H., Deng W., A note on Green’s relations in the semigroups T(X, ρ), Semigroup Forum, 2009, 79(1), 210–213
Sullivan R.P., Semigroups of linear transformations with restricted range, Bull. Austral. Math. Soc., 2008, 77(3), 441–453
Sullivan R.P., Semigroups of linear transformations with restricted kernel (submitted)
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Mendes-Gonçalves, S., Sullivan, R.P. Semigroups of transformations restricted by an equivalence. centr.eur.j.math. 8, 1120–1131 (2010). https://doi.org/10.2478/s11533-010-0066-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-010-0066-8