Abstract

For each subchain X′ of a chain X, let T<inf>RE</inf>(X,X′) denote the semigroup under composition of all full regressive transformations, α:X→X′ satisfying xα≤x for all x∈X. Necessary and sufficient conditions for T<inf>RE</inf>(X,X′) and T<inf>RE</inf>(Y,Y′) to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations T<inf>RE</inf>(X,X′) where X is one of several familiar subchains of ℝ, the chain of real numbers. © 2012 Springer Science+Business Media New York.