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Mathematics 2018, 6(8), 134; https://doi.org/10.3390/math6080134

Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
*
Author to whom correspondence should be addressed.
Received: 4 June 2018 / Revised: 9 July 2018 / Accepted: 2 August 2018 / Published: 4 August 2018
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Abstract

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)ρ, then (aα,bα)ρ) and R (if rR, then rαR). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively. View Full-Text
Keywords: transformation semigroup; Green’s relations; equivalence relation; cross-section transformation semigroup; Green’s relations; equivalence relation; cross-section
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Pookpienlert, C.; Honyam, P.; Sanwong, J. Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section. Mathematics 2018, 6, 134.

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