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.

翻译：我们研究了带乘法的Weihrauch格的等式理论，即由变量、格运算$\sqcup$、$\sqcap$、乘积$\times$以及有限并行化$(-)^*$构成的项之间的等式集合，这些等式在任意用Weihrauch度替换变量时均成立。我们基于有限图之间的可归约性给出了这些等式的组合刻画，进一步证明了在此意义下判定等式是否成立是多项式层次第三层的完全问题。