  The set of associative and commutative hypercomplex numbers, called the
perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient
conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices
are reviewed. The zero set is defined for non-invertible hypercomplex numbers
in a given PHA, and a characteristic function is proposed for calculating zero
set. Then PHA of different dimensions are considered. First, $2$-dimensional
PHAs are considered as examples to calculate their zero sets etc. Second, all
the $3$-dimensional PHAs are obtained and the corresponding zero sets are
investigated. Third, $4$-dimensional or even higher dimensional PHAs are also
considered. Finally, matrices over pre-assigned PHA, called perfect
hypercomplex matrices (PHMs) are considered. Their properties are also
investigated.
