Green's Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation

Let ${\cal T}_X$ be the full transformation semigroup on the set $X$, \ T_{E}(X)=\{f\in {\cal T}_X\colon \ \forall(a,b)\in E,(f(a),f(b))\in E\} \] be the subsemigroup of ${\cal T}_X$ determined by an equivalence $E$ on $X$. In this paper the set $X$ under consideration is a totally ordered set with $mn$ points where $m\geq 2$ and $n\geq 3$. The equivalence $E$ has $m$ classes each of which contains $n$ consecutive points. The set of all order preserving transformations in $T_{E}(X)$ forms a subsemigroup of $T_E(X)$ denoted by \ {\cal O}_{E}(X)=\{f\in T_{E}(X)\colon \ \forall\, x, y\in X, \ x\leq y \mbox{ implies } f(x)\leq f(y)\}. \] The nature of regular elements in ${\cal O}_{E}(X)$ is described and the Green's equivalences on ${\cal O}_{E}(X)$ are characterized completely.