The mathematical foundations of the cell method

Of special importance for the philosophy of the Cell Method (CM) is the classification diagram of the physical variables. Originally, the classification diagram was obtained on the basis of physical considerations on the associations between physical variables and geometry. We will show in this paper that we may obtain the same associations on the basis of mathematical considerations, thus deepening the mathematical foundations of the CM. This will allow us to recognize in the classification diagram of the Cell Method a structure of bialgebra, where the operators are generated by the outer product of the geometric algebra and the exterior product of the dual algebra of the enclosed exterior algebra. In doing so, the classification itself of the physical variables will take on a deeper meaning, by allowing us to associate the configuration variables with the geometric interpretation for the elements of a vector space and the source variables with the geometric interpretation for the elements of the dual vector space in the bialgebra. We will also discuss a new four-dimensional space/time cell-complex for studying time dependent phenomena with the CM.