Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence

Let TX be the full transformation semigroup on a set X, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}$$\end{document} be the subsemigroup of TX determined by an equivalence E on X. In this paper the set X under consideration is a totally ordered set with n points. The set of all order preserving transformations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{E^{*}}(X)$\end{document} forms 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}$T_{E^{*}}(X)$\end{document} denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_{E^*}(X)=\{\alpha\in T_{E^*}(X): \forall x,y\in X, x\leq y\Rightarrow x\alpha\leq y\alpha\}.$$\end{document} In this paper, we discuss Green’s relations for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O_{E^{*}}(X)$\end{document} and prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O_{E^{*}}(X)$\end{document} is a regular semigroup.