Isomorphisms of lexicographic powers of the reals

Abstract:

Given a chain $\Gamma$, we consider the lexicographic order ${\mathbb {R}^\Gamma }$. If $\Delta$ is a chain such that ${\mathbb {R}^\Gamma } \simeq {\mathbb {R}^\Delta }$, we examine the question of whether necessarily $\Gamma \simeq \Delta$. Under an additional hypothesis, we show that $\Gamma$ and $\Delta$ will have the same order types of well ordered subsets. Among other things, this yields an affirmative answer to the above question in the case where $\Gamma$ and $\Delta$ are ordinals.