Finding necessary and sufficient conditions for isomorphism between two semigroups of order-preserving transformations over an infinite domain with restricted range was an open problem in cite{FHQS}. In this paper, we show a proof strategy to answer that question.
For each subchain $X$ of a chain $X$, let $T_{RE}(X, X)$ denote the semigroup under composition of all full regressive transformations, $alpha:Xrightarrow X$ satisfying $xalphaleq x$ for all $xin X$. Necessary and sufficient conditions for $T_{RE}(X,
X)$ and $T_{RE}(Y,Y)$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X)$ where $X$ are familiar subchains of $R$, the chain of real numbers.
