An analysis of Ruspini partitions in Godel logic

By a Ruspini partition we mean a finite family of fuzzy sets {f(1,) ... ,f(n)} f(i) : [0,1] -> [0, 1]. such that Sigma(n)(i-1)f(i)(x) = 1 for all x is an element of [0, 1], where [0, 1] denotes the real unit interval. We analyze such partitions in the language of Godel logic. Our first main result identifies the precise degree to which the Ruspini condition is expressible in this language, and yields inter alia a constructive procedure to axiomatize a given Ruspini partition by a theory in Godel logic. Our second main result extends this analysis to Ruspini partitions fulfilling the natural additional condition that each f(i) has at most one left and one right neighbour, meaning that min(x is an element of,0.1) {fi(1) (x), fi(2) (x), fi(3) (x)} = 0 holds for i(1) not equal i(2) not equal i(3). (C) 2009 Elsevier Inc. All rights reserved.