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Regularity and Green's Relations for Semigroups of Transformations Preserving Orientation and an Equivalence

Authors:

Institution:

(1) School of Sciences, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, P. R. China;(2) Department of Mathematics, Xinyang Normal University, Xinyang, Henan, 464000, P. R. China

Abstract:

Let ${\cal T}_X$ be the full transformation semigroup on a set X. For a non-trivial equivalence E on X, let

$T_E (X) =\{ f\in {\cal T}_X \colon \ \forall \, (x,y)\in E,\, (f(x),f(y))\in E \}.$

Then T_E(X) is a subsemigroup of ${\cal T}_ X$ . For a finite totally ordered set X and a convex equivalence E on X, the set of all orientation-preserving transformations in T_E(X) forms a subsemigroup of T_E(X) which is denoted by OP_E(X). In this paper, under the hypothesis that the set X is a totally ordered set with mn (m ≥ 2,n ≥ 2) points and the equivalence E has m classes each of which contains n consecutive points, we discuss the regularity of elements and the Green's relations for OP_E(X).

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