  We develop the description of non-commutative geometry of the 4-dimensional
quantum Hall fluid's theory proposed recently by Zhang and Hu. The
non-commutative structure of fuzzy $S^{4}$ appears naturally in this theory.
The fuzzy monopole harmonics, which are the essential elements in this
non-commutative geometry, are explicitly constructed and their obeying the
matrix algebra is obtained. This matrix algebra is associative. We also propose
a fusion scheme of the fuzzy monopole harmonics of the coupling system from
those of the subsystems, and determine the fusion rule in such fusion scheme.
By products, we provide some essential ingredients of the theory of SO(5)
angular momentum. In particular, the explicit expression of the coupling
coefficients, in the theory of SO(5) angular momentum, are given. It is
discussed that some possible applications of our results to the 4-dimensional
quantum Hall system and the matrix brane construction in M-theory.
