Triangle functions generated by products of quantales

Abstract: It is known that, given a left continuous t-norm $T$ on $[0,1]$ and a right continuous $\CL$-operation $L$ on $[0,\infty]$, then their tensor product $L\otimes T$ is a triangle function on $\Delp$. Hence, $(\Delp,L\otimes T)$ becomes a partially ordered monoid. In this paper, it is shown that: (1) $L\otimes T$ coincides with the triangle function $\tau_{T,L}$ if and only if $L$ satisfies the condition $(LCS)$; (2) let $\CDp$ be the set of all non-defective distance distribution functions, then $(\CDp,L\otimes T)$ is a submonoid of $(\Delp,L\otimes T)$ if and only if $L$ has no zero divisor; (3) let $\CDp_c$ be the set of all continuous distance distribution functions, then for each continuous t-norm $T$, $(\CDp_c, L\otimes T)$ is a subsemigroup of $(\Delp,L\otimes T)$ if and only if $L$ satisfies the condition $(LS)$. Moreover, $(\CDp_c,L\otimes T)$ is an ideal of $(\CDp,L\otimes T)$ if and only if $L$ satisfies the cancellation law.