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Cauchy's equation on a space of distribution functions
Abstract
Let $\Delta\sp+$ be the space of probability distribution functions of nonnegative random variables, partially ordered by the usual pointwise ordering of functions, and let $\tau$ be a semigroup (triangle function) on $\Delta\sp+$. We say that $\varphi$: $\Delta\sp+\to\Delta\sp+$ satisfies Cauchy's equation for $\tau$, if $\varphi$($\tau$(F,G)) = $\tau$($\varphi$(F),$\varphi$(G)), for all F,G in $\Delta\sp+$; and we say that $\varphi$ is sup-continuous, if $\varphi$(sup$\sb{j\epsilon}\sb{J}F\sb{j}$) = sup $\sb{j\epsilon}\sb{J}\varphi$($F\sb{j}$), whenever $F\sb{j}$ is in $\Delta\sp+$, for all j in some index set J. We first use some recent lattice-theoretic results due to R. C. Powers (Order automorphisms of spaces of nondecreasing functions. J. Math. Anal. Appl. 136 (1988), 112-123.), to find all order automorphism solutions of Cauchy's equation for triangle functions of the form $\tau\sb{T,L}$ and $\tau\sb{T\sp\*,L}$ where T is a continuous Archimedean t-norm, $T\sp\*$ is a continuous Archimedean t-conorm and L is a binary operation on $R\sp+$ isomorphic to addition or $Max$(x + y $-$ 1,0). Next we give a representation of all sup-continuous solutions of Cauchy's equation for $\tau\sb{T,L}$. We conclude with some additional results and open problems.
Subject Area
Mathematics
Recommended Citation
Riedel, Thomas, "Cauchy's equation on a space of distribution functions" (1990). Doctoral Dissertations Available from Proquest. AAI9100532.
https://scholarworks.umass.edu/dissertations/AAI9100532