Weihrauch degrees without roots

TL;DR

This paper investigates the structure of Weihrauch degrees, demonstrating the existence of uncountably many degrees without roots and showing that certain omniscience principles lack roots, thus advancing the understanding of their algebraic properties.

Contribution

The paper proves the existence of uncountably many Weihrauch degrees without roots and establishes that LPO and LLPO do not have roots, answering a question by Arno Pauly.

Findings

Uncountably many pairwise incomparable degrees without roots.

LPO and LLPO omniscience principles lack roots.

Provides new insights into the algebraic structure of Weihrauch degrees.

Abstract

We answer the following question by Arno Pauly: "Is there a square-root operator on the Weihrauch degrees?". In fact, we show that there are uncountably many pairwise incomparable Weihrauch degrees without any roots. We also prove that the omniscience principles of LPO and LLPO do not have roots.