DEGENERATE AND NONDEGENERATE CONVEX DECOMPOSITION OF FINITE FUZZY PARTITIONS .1.

In Bezdek and Harris (1979) an algorithm (called MiniMax, shortly MM algorithm) for the convex decomposition of a fuzzy partition has been proposed. In this paper another decomposition algorithm (called MiniMiniMax, shortly MMM algorithm) is considered. A comparative study of these algorithms is done. From this study we may conclude that (i) the MM convex decomposition sequence is not lexicographically larger than any other convex decomposition; (ii) the conjecture from Bezdek and Harris (1979) concerning the length of the MM decomposition fails. Some properties of the spaces of fuzzy partitions are also given. In Part II a theorem which states a necessary and sufficient condition for the existence of non-degenerate convex decompositions is proved. An algorithm for non-degenerate convex decomposition of a fuzzy partition inspired by the constructive proof of this theorem is proposed. This algorithm builds a positive path through the matrix representing a fuzzy partition. The convergence of the algorithm and the monotony of the coefficients sequence in the convex decomposition are proved. Other two results (the limited cardinality and the heredity property) concerning this algorithm are also given.