# On the algebraic structure of Weihrauch degrees

arXiv: https://arxiv.org/abs/1604.08348 · HTML: https://mlsanity.com/paper/1604.08348/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2016.

## Built on by (most cited first, 2 of 2)

- [On the Complexity of Computing G"odel Numbers](https://mlsanity.com/paper/2302.04213/) (2023)
- [Uniform Computability of PAC Learning](https://mlsanity.com/paper/2601.18663/) (2026)

## Abstract (arXiv)

We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison with similar structures such as residuated lattices and concurrent Kleene algebras. Introducing the notion of an ideal with respect to the compositional product, we can consider suitable quotients of the Weihrauch degrees. We also prove some specific characterizations using the implication. In order to introduce and study compositional products and implications, we introduce and study a function space of multi-valued continuous functions. This space turns out to be particularly well-behaved for effectively traceable spaces that are closely related to admissibly represented spaces.