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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

Citations:

Zbl 0793.06006
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