On the algebraic structure of Weihrauch degrees

TL;DR

This paper explores the algebraic structure of Weihrauch degrees by introducing new operations, studying their properties, and comparing them to other algebraic systems, thereby deepening understanding of their mathematical framework.

Contribution

It introduces compositional products and implications on Weihrauch degrees, analyzes their algebraic properties, and develops a function space framework for multi-valued continuous functions.

Findings

Distributivity laws are characterized within the algebraic structure.

Quotients of Weihrauch degrees are constructed via ideals.

A well-behaved function space for multi-valued continuous functions is developed.

Abstract

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.