The equational theory of the Weihrauch lattice with multiplication

TL;DR

This paper explores the algebraic structure of the Weihrauch lattice with multiplication, providing a combinatorial characterization and complexity results for its equational theory.

Contribution

It offers a combinatorial description of the equational theory and proves its decision problem is complete for the third level of the polynomial hierarchy.

Findings

Equational theory characterized by reducibility between finite graphs

Deciding equations is complete for the third level of the polynomial hierarchy

Provides a new algebraic perspective on Weihrauch degrees

Abstract

We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite parallelization $(-)^*$ which are true however we substitute Weihrauch degrees for the variables. We provide a combinatorial description of these in terms of a reducibility between finite graphs, and moreover, show that deciding which equations are true in this sense is complete for the third level of the polynomial hierarchy.