On the Arithmetic of Lexicographic Exponentiation
Salma Kuhlmann · 2002
In [H] Hausdorff developed several arithmetic operations on totally ordered sets, generalizing Can-tor’s ordinal arithmetic. Many open questions arise naturally, that we have been studying in the last few years. This talk will give an overview of our main results: In [K], we studied lexicographic powers of the form RΓ, and asked whether the exponent is an isomorphism invariant: Theorem 1 Let α be an ordinal, and J a chain in which the chain R does not embed. Assume that ϕ is an embedding of Rα in RJ. Then α embeds in J. In particular, if α and β are distinct ordinals, then the chains Rα and Rβ are nonisomorphic. This theorem is used in [W] to classify the convex congruences of such powers. On the other hand, after establishing further arithmetic rules, we study in [HKM] nonisomorphic chains for which the corresponding lexicographic powers are isomorphic: for a countable infinite ordinal α, we show that Rα ∗+α and Rα are isomorphic. We show that RR and RQ are nonisomorphic. We show that ∆R is 2-homogeneous, where ∆ is a countable ordinal ≥ 2. Further related open questions arise while studying the question of defining an exponential function on a power series field: in [KKS2] we study convex embeddings of a chain Γ in a lexicographic power ∆Γ and prove