Equations over Finite Monoids with Infinite Promises

Larrauri and \v{Z}ivn\'y [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.