Extending ordinal exponentiation to non‑trichotomous bases via quotiented lists

Determine whether ordinal exponentiation α^β in homotopy type theory can be defined for base ordinals α that do not possess a trichotomous least element by constructing a quotiented‑list representation of decreasing lists, thereby fusing the abstract supremum‑based and the concrete decreasing‑list approaches into a unified definition applicable to arbitrary base ordinals.

Background

The paper presents two constructive definitions of ordinal exponentiation in homotopy type theory: an abstract definition using suprema and transfinite recursion, and a concrete definition using decreasing lists inspired by Sierpiński’s finite-support functions. These are shown to be equivalent when the base ordinal has a trichotomous least element, enabling transfer of algebraic and decidability properties.

However, the concrete decreasing-list construction relies on the base ordinal having a trichotomous least element to ensure the list type forms an ordinal. The abstract construction is more general but still leverages positivity of the base in its specification. Extending the concrete approach beyond the trichotomous assumption—potentially by quotienting lists to enforce appropriate identifications—would broaden applicability of the unified exponentiation to bases lacking trichotomy.