Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions

Abstract

The complete-lattice tensor product $\mathbb{E}=[0,1]\otimes[0,1]$ 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 \[ (f\otimes_{_{S,T}}g)(x)=\sup_{S(u,v)=x}T(f(u),g(v)) \] on $\mathbb{E}$. 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 $\otimes_{_{S,T}}$ is a t-conorm on $\mathbb{E}$ 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 $\mathbb{E}$, 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 $L\neq\max$, continuity of $τ_{_{T^*,L}}$ is equivalent to continuity of $T^*$. This resolves the remaining $τ_{_{T^*,L}}$ part of the Schweizer--Sklar problem.