A Classification of Regressive Transformation Semigroups on Chains

TL;DR

This paper classifies regressive transformation semigroups on chains, providing conditions for their isomorphism and applying these results to familiar real number subchains.

Contribution

It offers a comprehensive classification of regressive transformation semigroups on chains, including isomorphism conditions and applications to real number subchains.

Findings

Necessary and sufficient conditions for semigroup isomorphism.

Classification of regressive transformation semigroups on real subchains.

Application of isomorphism theorem to familiar chains.

Abstract

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.