Foundations of Fuzzy Logic and Semantic Web Languages

116 Foundations of Fuzzy Logic and Semantic Web Languages

Proposition 35 (Mostert–Shields t-norm characterization [241]). The fol-

lowing are equivalent:

1. ⊗ is a continuous t-norm;

2. ⊗ is uniquely representable as an ordinal sum of continuous Archimedean

t-norms;

3. there is a uniquely determined countable family ((a

α

, b

α

))

α∈A

of non-

empty, pairwise disjoint open sub-intervals of [0, 1] such that

(a) if ⊗ is strict on I = (a

α

, b

α

) then ⊗

|I

is isomorphic to product

t-norm ⊗

p

;

(b) if ⊗ is not strict, i.e., nilpotent, on I = (a

α

, b

α

) then ⊗

|I

is iso-

morphic to Lukasiewicz t-norm ⊗

l

;

(c) if a, b ∈ [0, 1] are such that there is no I = (a

α

, b

α

) such that