| Abstract: | We introduce a frame cellular automaton as a broad generalization of an earlier study on quasigroup-defined cellular automata. It consists of a triple (F,R,EF) where, for a given finite set X of cells, the frame F is a family of subsets of X (called elementary frames, denoted Si, i=1,…,n), which is a cover of X. A matching configuration is a map  which attributes to each cell a state in a finite set G under restriction of a set of local rules R={Ri∣i=1,…n}, where Ri holds in the elementary frame Si and is determined by an (|Si|-1)-ary quasigroup over G. The frame associated map EF models how a matching configuration can be grown iteratively from a certain initial cell-set. General properties of frames and related matroids are investigated. A generating set S⊂X is a set of cells such that there is a bijection between the collection of matching configurations and GS. It is shown that for certain frames, the algebraically defined generating sets are bases of a related geometric-combinatorially defined matroid. |
