Functorial equations for lexicographic products

Abstract:

We generalize the main result of an earlier paper by the authors (Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177–3183) concerning the convex embeddings of a chain $\Gamma$ in a lexicographic power $\Delta ^{\Gamma }$. For a fixed non-empty chain $\Delta$, we derive necessary and sufficient conditions for the existence of non-empty solutions $\Gamma$ to each of the lexicographic functorial equations \[ (\Delta ^{\Gamma })^{\leq 0} \simeq \Gamma \>, \ \ (\Delta ^{\Gamma }) \simeq \Gamma \ \ \mbox { and } \ \ (\Delta ^{\Gamma })^{<0} \simeq \Gamma \;.\]