9 The theory of chi processes with the mismatch operator is studied. Four congruence relations are investigated. These are late open congruence, early open congruence, ground congruence and 11 barbed congruence. The late and early open congruence relations are the chi calculus counterparts of the weak late and early congruence relations of pi calculus. Both turn out to be special cases 13 of the ground congruence and the barbed congruence. The ground congruence is essentially the open congruence. Complete systems are given for all the four congruence relations. These 15 systems use some interesting tau laws unknown from previous studies of the chi calculus without the mismatch combinator. The results of this paper point out that the mismatch operator changes 17 the algebraic semantics of chi calculus dramatically. They also correct some common mistakes in literature. 19 c © 2002 Published by Elsevier Science B.V.