it seems to be reasonable to consider the following generalization of cellular automata rules. Let us consider a square matrix lines (rows and columns) in place of an automaton cells. Furthermore, let rules for changing a value of given cell (here, a line) are elementary matrix operations like switching lines.
In a pre-print https://arxiv.org/abs/1705.04376, I introduced a model of matrices based on these assumptions for a simple relation of order (which could be understood as some kind of relation which puts some abstract objects in order). As a consequence, eigenvectors for these matrices generate a set of three-dimensional orthogonal vectors which could represent three-dimensional physical objects with some property. An evolution of the relations matrix postulated here gives quantized values of a 'state' vector which I tried to incorporate.
My model starts from reasonable claim of an existence of a simple relation between simple objects. As a consequence, it reveals some simple characteristics, like three dimensional sub-spaces or a quantized spectrum of values. Its compatibility to quantum mechanics description is still hypothetical but coincidences with a quantum description are promising.
I would be thankful for all your comments.