Triangle functions generated by products of quantales

TL;DR

This paper explores how tensor products of triangular norms and conorms induce triangle functions on distance distribution functions, establishing conditions for monoid and subsemigroup structures.

Contribution

It characterizes when tensor products of t-norms and t-conorms produce triangle functions and algebraic structures on sets of distance distribution functions.

Findings

Triangle functions induced by tensor products are characterized by continuity and property conditions.

Submonoid and subsemigroup structures depend on properties like no zero divisors and continuity.

Conditions for ideals involve adherence to the cancellation law.

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 $(\Delta^+, L \otimes T)$. The main results are as follows: (1) if $L$ is continuous, then $\tau_{T,L}$ is a triangle function on $\Delp$ if and only if $\tau_{T,L}=L\otimes T$, 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.