Answering an open problem on t-norms for type-2 fuzzy sets

This paper proves that a binary operation ⋆ on [0, 1], ensuring that the binary operation ⋎ is a t-norm or ⋏ is a t-conorm, is a t-norm, where ⋎ and ⋏ are special convolution operations defined by (f ⋏ g)(x) = sup {f (y) ★ g(z) : y △ z = x}, (f ⋎ g)(x) = sup {f (y) ★ g(z) : y ▽ z = x}, for any f, g ∈ Map([0, 1], [0, 1]), where △ and ▽ are a continuous t-norm and a continuous t-conorm on [0, 1], answering negatively an open problem posed in [8]. Besides, some characteristics of t-norm and t-conorm are obtained in terms of the binary operations ⋎ and ⋏.