Triangle functions generated by products of quantales

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.