We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations , , the product ×, 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.
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Neumann et al. (2024) studied this question.
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