Larrauri and Živný [ICALP'25/ACM ToCL'24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both monoids are finite and $M$ admits a homomorphism to $N$. Using the algebraic approach to promise constraint satisfaction problems, we extend their complexity classification in two directions: we obtain a complexity dichotomy in the case where arbitrary relations are added to the monoids, and we moreover allow the monoid $M$ to be finitely generated.

翻译：拉劳里和日夫尼[ICALP'25/ACM ToCL'24]最近建立了在假设解存在于幺半群$M$的前提下，求解幺半群$N$上方程组问题的完整复杂度分类，其中两个幺半群均为有限且$M$具有到$N$的同态。利用承诺约束满足问题的代数方法，我们在两个方向上扩展了他们的复杂度分类：在向幺半群添加任意关系的条件下得到了复杂度二分法，并且进一步允许$M$为有限生成幺半群。