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

In Bezdek and Harris (1979) an algorithm (called MiniMax, shortly MM a lgorithm) for the convex decomposition of a fuzzy partition has been p roposed. In this paper another decomposition algorithm (called MiniMin iMax, shortly MMM algorithm) is considered. A comparative study of the se algorithms is done. From this study we may conclude that (i) the MM convex decomposition sequence is not lexicographically larger than an y other convex decomposition; (ii) the conjecture from Bezdek and Harr is (1979) concerning the length of the MM decomposition fails. Some pr operties of the spaces of fuzzy partitions are also given. In Part II a theorem which states a necessary and sufficient condition for the ex istence of non-degenerate convex decompositions is proved. An algorith m for non-degenerate convex decomposition of a fuzzy partition inspire d by the constructive proof of this theorem is proposed. This algorith m builds a positive path through the matrix representing a fuzzy parti tion. The convergence of the algorithm and the monotony of the coeffic ients sequence in the convex decomposition are proved. Other two resul ts (the limited cardinality and the heredity property) concerning this algorithm are also given.