STRUCTURE OF DETERMINATIVE SUBSPACE IN TRIANGULAR CELL SPACE : INFORMATION SCIENCE APPROACH TO BIOMATHEMATICS, VIII
Masako Yamaguchi · Bulletin of Mathematical Statistics · 1971
The notion of stable configuration in a Sn' cell space which is invariant under any application of local majority transformation (LMT) was introduced in Kitagawa and Yamaguchi [1] and the notion of determinative subspace in a 4(71) cell space which determines a structure of a stable configuration was introduced in our paper [2]. Some structual properties of determinative subspace in ZI(n) cell space were suggested throughout various examples of determinative subspace in Kitagawa and Yamaguchi [3]. According to the definitions of generative and non-generative determinative subspace in 4(n) cell space given by Kitagawa [4] in view of propagation of determined cells, these examples are mostly concerned with those of generative determinative subspace in 4(n) cell space. The purpose of this paper is to give a deeper investigation for the constructions of determinative subspaces in 4(n) cell space. In SECTION 2 we shall introduce several notions which are indispensable for a construction procedure of generative determinative subspace in 4(m) cell space such as a convex set, a spiny convex set and a two-cell extension of a set and so on. In SECTION 3 we shall give a certain type of construction procedure of any generative determinative subspace in zl(n) cell space. In APPENDIX we shall give an example of our construction process of a generative determinative subspace obtained by using this proceduce. In SECTION 4 we shall introduce several notions of elementary subsets and superposition of elementary subsets and decomposition of the whole cell space into a family of superposed elementary subsets. These nations are fundamental tools for proving THEOREM 3 and 4, which give us a construction procedure and hence a structural characteristic feature of any non-generative determinative subspace in 4(n) cell space. The last SECTION 5 is devoted to the another proof of THEOREM 2 in our previours paper [2] which appeals to LEMMA 4 in the present paper prepared for giving our construction procedure of non-generative determinative determinative