For each subchain $X'$ of a chain $X$, let $T_{ RE}(X, X')$ denote the semigroup under composition of all full regressive transformations, $\alpha:X\rightarrow X'$ satisfying $x\alpha\leq x$ for all $x\in X$. 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.
Categories: math.RA