Triangle functions generated by products of quantales
Hongliang Lai, Qingzhu Luo
TL;DR
The paper studies triangle functions on distance distribution functions $\\Delta^+$ generated by tensor products $L\\otimes T$ of a left-continuous t-norm $T$ on $[0,1]$ and a right-continuous t-conorm $L$ on $[0,\\infty]$, and analyzes when the classical construction $\\tau_{T,L}$ coincides with $L\\otimes T$. It proves that $\\tau_{T,L}$ is a triangle function on $\\Delta^+$ if and only if $\\tau_{T,L}=L\\otimes T$, which is equivalent to $L$ satisfying the property $(LCS)$ (and hence to a related continuity condition) when $T$ is left-continuous. The paper further provides sharp closure criteria for three natural subfamilies of distance distribution functions: $\\mathcal{D}^+$ is closed under $L\\otimes T$ iff $L$ has no zero divisors; $\\mathcal{D}^+_0$ is closed iff $T$ is continuous and $L$ satisfies $(LS)$; and, when $T$ is continuous, $\\mathcal{D}^+_c$ is closed iff $(LS)$ holds, with $\\mathcal{D}^+_c$ forming an ideal of $\\mathcal{D}^+$ under suitable conditions (including the cancellation law for $L$). These results sharpen and extend Schweizer–Saminger theory via a quantale/tensor-product framework, with implications for probabilistic metric spaces and ordered algebraic structures.
Abstract
This paper investigates triangle functions induced by tensor products of triangular norms and conorms. For any left continuous t-norm $T$ on $[0,1]$ and any right continuous t-conorm $L$ on $[0,\infty]$, the tensor product $L\otimes T$ induces a triangle function on $\Delp$, giving rise to a partially ordered monoid structure on $(Δ^+, L \otimes T)$. The main results are as follows: (1) if $L$ is continuous, then $τ_{T,L}$ is a triangle function on $\Delp$ if and only if $τ_{T,L}=L\otimes T$, which in turn holds if and only if $L$ satisfies the property (LCS); (2) for $\CDp$, the set of all non-defective distance distribution functions, $(\CDp,L\otimes T)$ forms a submonoid of $(\Delp,L\otimes T)$ if and only if $L$ has no zero divisors; (3)for $\CDp_c$, the set of all continuous distance distribution functions, if the t-norm $T$ is continuous, then $(\CDp_c,L\otimes T)$ is a subsemigroup of $(\Delp,L\otimes T)$ if and only if $L$ satisfies the property (LS). Furthermore, $(\CDp_c,L\otimes T)$ is an ideal of $(\CDp, L\otimes T)$ if and only if $L$ adheres to the cancellation law.