Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions
Abstract
The complete-lattice tensor product 𝔼=[0,1]⊗[0,1] arises naturally in the algebra of truth values for type-2 fuzzy sets. For a monotone surjective map S that is continuous on [0,1[2 and a monotone map T having $1$ as identity, we consider the generalized supremum convolution (f⊗S,Tg)(x)=supS(u,v)=xT(f(u),g(v)) on 𝔼. Rather than assuming classical algebraic structures for S and T from the outset, we ask when they are forced by requiring the induced convolution itself to be a t-conorm. We prove that ⊗S,T is a t-conorm on 𝔼 if and only if S is a continuous t-conorm and either S=max or T is a right-continuous t-norm; in the exceptional case S=max, the convolution is simply the pointwise maximum for every admissible T. The same convolution mechanism also yields an order-dual description of triangle functions on the space of distance distribution functions Δ+. Using an order-reversing isomorphism between Δ+ and a boundary variant of 𝔼, we show that τT*,L is a triangle function precisely when L is a continuous t-conorm with no zero divisor and, unless L=max, T* is a left-continuous t-conorm; moreover, when L≠max, continuity of τT*,L is equivalent to continuity of T*. This resolves the remaining τT*,L part of the Schweizer--Sklar problem.
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BibTeX
@misc{luo2026generalized,
title = {{Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions}},
author = {Mengyu Luo and Lili Shen and Jie Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40659},
eprint = {2610.40659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40659},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions
AU - Mengyu Luo
AU - Lili Shen
AU - Jie Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.40659
UR - https://arxiv.org/abs/2610.40659
AB - The complete-lattice tensor product 𝔼=[0,1]⊗[0,1] arises naturally in the algebra of truth values for type-2 fuzzy sets. For a monotone surjective map S that is continuous on [0,1[^2 and a monotone map T having $1$ as identity, we consider the generalized supremum convolution (f⊗__S,Tg)(x)=sup_S(u,v)=xT(f(u),g(v)) on 𝔼. Rather than assuming classical algebraic structures for S and T from the outset, we ask when they are forced by requiring the induced convolution itself to be a t-conorm. We prove that ⊗__S,T is a t-conorm on 𝔼 if and only if S is a continuous t-conorm and either S=max or T is a right-continuous t-norm; in the exceptional case S=max, the convolution is simply the pointwise maximum for every admissible T. The same convolution mechanism also yields an order-dual description of triangle functions on the space of distance distribution functions Δ^+. Using an order-reversing isomorphism between Δ^+ and a boundary variant of 𝔼, we show that τ__T^*,L is a triangle function precisely when L is a continuous t-conorm with no zero divisor and, unless L=max, T^* is a left-continuous t-conorm; moreover, when L≠max, continuity of τ__T^*,L is equivalent to continuity of T^*. This resolves the remaining τ__T^*,L part of the Schweizer--Sklar problem.
ER -