$F$-congruences on $F$-inverse monoids

We study the class of congruences on $F$-inverse monoids that respect the unary operation $a\mapsto \text{mx}a$, assigning to each element the maximum element of its $\sigma$-class, which we refer to as $F$-congruences. We characterize kernel normal systems and congruence pairs on $F$-inverse monoids which give rise to such congruences and establish a number of further related results including a classification of $F$-congruence-free $F$-inverse monoids.