There are at present a large number of theoretical and algorithmic results relating to one-variable polynomial matrices arising from one-dimensional multivariable systems. In recent years many of the theoretical results have been extended to two-variable polynomial matrices arising from two-dimensional multi variable systems, such as delay-differential or partial differential systems. However there has been no major attempt to extend the algorithmic results associated with single variable polynomial matrices to two-variable or multivariable polynomial matrices. This thesis investigates further some of the extensions of the algebra of one-dimensional multivariable systems to two-dimensional multivariable systems.