Congruence identities of regularized multiple zeta values involving a pair of index sets

Let Z(l)((n)) be the vector space spanned by multiple zeta values (MZVs) of weight l and depth less than n + 1. Let (Z) over bar ((n-1))(l) be the subspace spanned by MZVs of weight l and depth n-1, and let P-l((n)) be the subspace given by the intersection of Z(l)((n)) and the vector space spanned by products of MZVs. In the present paper, we prove congruence identities of regularized multiple zeta values involving a pair of index sets modulo (Z) over bar ((n-1))(l) + P-l((n)). We also obtain a proof of the parity result, and a congruence sum formula for MZVs.