The Pascal matroid as a home for generating sets of cellular automata configurations defined by quasigroups

Abstract

Consider a one-dimensional CA whose local evolution rule is defined by a quasigroup ( G,★). When the initial state is only known over a finite string of adjacent cells, the CA's orbit (state-time diagram) is only defined on a triangular array of cells. This well-defined triangular part of the orbit, in which each cell has a value in G, is called a ( G,★)-configuration. A generating set is a subset of the triangular array such that any attribution of G-values to the cells of this subset determines a unique ( G,★)-configuration. It turns out that the collection of all potential generating sets form a particular matroid, which we have called the Pascal matroid. The main subject of this work is related to the question whether every base of this matroid actually occurs as a generating set for a ( G,★)-configuration.