# Joins in the strong Weihrauch degrees

## TL;DR

This paper establishes that the strong Weihrauch degrees form a non-distributive lattice by introducing a join operation, contrasting with the distributive structure of Weihrauch degrees, and answers an open question in the field.

## Contribution

It proves that the strong Weihrauch degrees form a lattice with a join operation, and shows this lattice is not distributive, resolving an open problem.

## Key findings

- Strong Weihrauch degrees form a lattice with a join operation.
- The lattice of strong Weihrauch degrees is not distributive.
- Strong and Weihrauch degrees are not isomorphic structures.

## Abstract

The Weihrauch degrees and strong Weihrauch degrees are partially ordered structures representing degrees of unsolvability of various mathematical problems. Their study has been widely applied in computable analysis, complexity theory, and more recently, also in computable combinatorics. We answer an open question about the algebraic structure of the strong Weihrauch degrees, by exhibiting a join operation that turns these degrees into a lattice. Previously, the strong Weihrauch degrees were only known to form a lower semi-lattice. We then show that unlike the Weihrauch degrees, which are known to form a distributive lattice, the lattice of strong Weihrauch degrees is not distributive. Therefore, the two structures are not isomorphic.