A Classification of Regressive Transformation Semigroups on Chains

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

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