An analysis of Ruspini partitions in Gödel logic

Abstract

By a Ruspini partition we mean a finite family of fuzzy sets {f 1 , . . . , f n }, f i :, where [0, 1] denotes the real unit interval. We analyze such partitions in the language of Gödel 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 Gödel 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∈[0,1] {f i 1 (x), f i 2 (x), f i 3 (x)} = 0 holds for i 1 = i 2 = i 3 .

Valuations in Gödel Logic, and the Euler Characteristic

“…In various guises, the above notion of equivalent assignments plays a crucial rôle in the investigation of Gödel logic; see e.g. [5,2]. For our purposes here, we observe that distinct 2-valued (=Boolean) assignments are not equivalent, so that there are 2 n equivalence classes of such assignments over the first n variables.…”

Valuations in Nilpotent Minimum Logic

“…Conversely, for every p maximal prime filter in NM n there exists an NM algebras homomorphism h p : NM n → 3, induced by the natural quotient map NM n → NM n /p composed with the embedding NM n /p → 3 given by Lemma II.2. Thanks to the bijection established by (7) and (8), we are able to associate an assignment µ hp with every minimal idempotent join irreducible element p in NM n . And the Lemma is settled.…”

“…For background on Gödel logic see, e.g.,[15]. The characterization of Gödel algebra used in the cited papers is provided in[3],[8],[11].…”