A Classification of Regressive Transformation Semigroups on Chains

Abstract

For each subchain $X^{\prime }$ of a chain $X$, let $T_{RE}(X,X^{\prime })$ denote the semigroup under composition of all full regressive transformations, $\alpha :X\to X^{\prime }$ satisfying $x\alpha \le x$ for all $x\in X$. Necessary and sufficient conditions for $T_{RE}(X,X^{\prime })$ and $T_{RE}(Y,Y^{\prime })$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X^{\prime })$ where $X$ are familiar subchains of $\mathbb{R}$, the chain of real numbers.

To appear in Semigroup Forum