  We provide some language for algebraic study of the mapping class groups for
surfaces with non-connected boundary. As applications, we generalize our
previous results on Dehn twists to any compact connected oriented surfaces with
non-empty boundary. Moreover we embed the `smallest' Torelli group in the sense
of Putman into a pro-nilpotent group coming from the Goldman Lie algebra. The
graded quotients of the embedding equal the Johnson homomorphisms of all
degrees if the boundary is connected.
