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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal T}_X$\end{document} be the full transformation semigroup on a set X. For a non-trivial equivalence E on X, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_E (X) =\{ f\in {\cal T}_X \colon \ \forall \, (x,y)\in E,\, (f(x),f(y))\in E \}.$\end{document} Then TE(X) is a subsemigroup of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal T}_ X $\end{document}. For a finite totally ordered set X and a convex equivalence E on X, the set of all orientation-preserving transformations in TE(X) forms a subsemigroup of TE(X) which is denoted by OPE(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 OPE(X).