  For further investigating the underlying structures of the D=3, N=4
Chern-Simons-matter (CSM) theories, we suggest a new concept and procedure for
"fusing" two superalgebras into a single new superalgebra. The starting
superalgebras may be those used in the previous construction of the
double-symplectic 3-algebras in the N=4 CSM theories: The bosonic parts of
these two superalgebras share at least one simple factor or U(1) factor. We are
able to provide two different methods to do the "fusion". Several explicit
examples are presented to demonstrate the "fusion" procedure. We also
generalize the "fusion" procedure so that more than two superaglebras can be
fused into a single one, provided some conditions are satisfied. It is shown
that two or more N=4 theories with different gauge groups may be associated
with the same "fused" superalgebra.
