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

Chollawat Pookpienlert, Preeyanuch Honyam and Jintana Sanwong
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Chollawat Pookpienlert: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Preeyanuch Honyam: Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Jintana Sanwong: Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Mathematics, 2018, vol. 6, issue 8, 1-12

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 ( a α , b α ) ∈ ρ ) and R (if r ∈ R , then r α ∈ R ). 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.

Keywords: transformation semigroup; Green’s relations; equivalence relation; cross-section (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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