  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.
