  This paper gives a systematic study of matching dialgebras corresponding to
the operad $As^{(2)}$ in \cite{Zi} as the only Koszul self dual operad there
other than the operads of associative algebras and Poisson algebras. The close
relationship of matching dialgebras with semi-homomorphisms and matched pairs
of associative algebras are established. By anti-symmetrizing, matching
dialgerbas are also shown to give compatible Lie algebras, pre-Lie algebras and
PostLie algebras. By the rewriting method, the operad of matching dialgebras is
shown to be Koszul and the free objects are constructed in terms of tensor
algebras. The operadic complex computing the homology of the matching
dialgebras is made explicit.
