## On sup-continuous triangle functions.(English)Zbl 0802.60022

Let $$\Delta^ +$$ denote the set of probability distribution functions of nonnegative random variables, partially ordered by the pointwise partial ordering of functions. Let $$\Delta^ +_ \delta$$ consist of the functions $$\delta_{a,b}$$ in $$\Delta^ +$$ which are zero up to $$a$$ and $$b$$ thereafter. A triangle function $$\tau$$ is a partially ordered semigroup on $$\Delta^ +$$, with $$\delta_{0,0}= \varepsilon_ \infty$$ as zero and $$\delta_{0,1}= \varepsilon_ 0$$ as identity. A triangle function is sup-continuous if it preserves arbitrary suprema in each variable.
We use a lattice-theoretic result of R. C. Powers [Order 7, No. 1, 83-96 (1990; Zbl 0793.06006)] to characterize the set of sup-continuous triangle functions which map $$\Delta^ +_ \delta \times \Delta^ +_ \delta$$ into $$\Delta^ +_ \delta$$. These are the triangle functions of the form $\tau_{T,L} (F,G) (x)= \sup \{T (F(u), G(v))\mid L(u,v)= x\},$ for all $$F$$, $$G$$ in $$\Delta^ +$$, $$x$$ in $$R^ +$$, where $$T$$ is a $$t$$-norm, i.e., an ordered semigroup on $$[0,1]$$ having 0 as zero and 1 as identity (e.g. product), and $$L$$ is a semigroup on $$R^ +$$, with 0 as identity and $$\infty$$ as zero (e.g. addition).

### MSC:

 60E99 Distribution theory 39B99 Functional equations and inequalities 06F05 Ordered semigroups and monoids

Zbl 0793.06006
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