The definition of open bisimilarity on the χ-processes does not give rise to a sensible relation on the χ-processes with the mismatch operator. The paper proposes ground open congruence as a principal open congruence on the χ-processes with the mismatch operator. Thealgebraic properties of the ground congruence is studied. The paper also takes a close look at barbed congruence. This relation is similar to the ground congruence. The precise relationship between the two is worked out. It is pointed out that the sound and complete system for the ground congruence can be obtained by removing one tau law from the complete system for the barbed congruence.