  We present a systematic algebraic approach for the weak coupling of Cauchy
problems to multiple Lohe tensor models. For this, we identify an admissible
Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol
consisting of four tuples in terms of a size vector, a natural frequency
tensor, a coupling strength tensor and admissible initial configuration. In
this way, the collection of all admissible Cauchy problems to the LT models is
equivalent to the space of characteristic symbols. On the other hand, we
introduce a binary operation, namely "{\it fusion operation}" as a binary
operation between the characteristic symbols. It turns out that the fusion
operation satisfies the associativity and admits the identity element in the
space of characteristic symbols which naturally forms a monoid. By virtue of
the fusion operation, the weakly coupled system of multiple LT models can be
obtained by applying the fusion operation of multiple characteristic symbols
corresponding to the LT models. As a concrete example, we consider a weak
coupling of the swarm sphere model and the Lohe matrix model, and provide a
sufficient framework leading to emergent dynamics to the proposed weakly
coupled model.
