Chains and antichains in the Weihrauch lattice

TL;DR

This paper investigates the structure of the Weihrauch lattice, focusing on the existence, bounds, and extendibility of long chains and antichains, and relates some properties to the continuum hypothesis.

Contribution

It characterizes when uncountable chains have upper bounds, proves no cofinal chains exist, and links the existence of certain sequences to the continuum hypothesis.

Findings

No cofinal chains of any order type exist in the Weihrauch degrees.

Existence of coinitial sequences of non-zero degrees is equivalent to CH.

Provides necessary conditions for the maximality of antichains.

Abstract

We study the existence and the distribution of "long" chains in the Weihrauch degrees, mostly focusing on chains with uncountable cofinality. We characterize when such chains have an upper bound and prove that there are no cofinal chains (of any order type) in the Weihrauch degrees. Furthermore, we show that the existence of coinitial sequences of non-zero degrees is equivalent to $\mathrm{CH}$. Finally, we explore the extendibility of antichains, providing some necessary conditions for maximality.