Triangle functions generated by products of quantales

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 $[0,\mathrm{\infty }]$, the tensor product $L\otimes T$ induces a triangle function on $\text{Delp}$, giving rise to a partially ordered monoid structure on $({\mathrm{\Delta }}^{+},L\otimes T)$. The main results are as follows: (1) if $L$ is continuous, then ${\tau }_{T,L}$ is a triangle function on $\text{Delp}$ 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 $\text{CDp}$, the set of all non-defective distance distribution functions, $(\text{CDp},L\otimes T)$ forms a submonoid of $(\text{Delp},L\otimes T)$ if and only if $L$ has no zero divisors; (3)for ${\text{CDp}}_c$, the set of all continuous distance distribution functions, if the t-norm $T$ is continuous, then $({\text{CDp}}_c,L\otimes T)$ is a subsemigroup of $(\text{Delp},L\otimes T)$ if and only if $L$ satisfies the property (LS). Furthermore, $({\text{CDp}}_c,L\otimes T)$ is an ideal of $(\text{CDp},L\otimes T)$ if and only if $L$ adheres to the cancellation law.