Matrix multiplication for finite algebraic systems

Finite algebraic systems having an arbitrary multiplication operator and an associative, communicative addition operator are investigated. It is shown that, for each such algebraic system, matrix multiplication is linear-time reducible to integer matrix multiplication. A consequence is that any fast algorithm for integer matrix multiplication can be converted into a fast algorithm for matrix multiplication. The reduction is applicable to sequential computation and can be extended to parallel computation. Of interest are the issues involved in performing matrix multiplication over a finite system where addition is associative and commutative, in an efficient single-instruction streams, multiple-data streams (SIMD) manner. It is shown that an SIMD integer matrix multiplication algorithm can be converted into an SIMD matrix multiplication algorithm for such an algebraic system. These reductions are applied to matrix closure where the finite algebraic system is a closed semiring.