Perfect hypercomplex algebras: Semi-tensor product approach

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $2$-dimensional PHAs are investigated. Second, all the $3$-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $4$- and higher dimensional PHAs are also considered.