Kybernetika 54 no. 2, 268-278, 2018

On $\star$-associated comonotone functions

Ondrej Hutník and Jozef PócsDOI: 10.14736/kyb-2018-2-0268

Abstract:

We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper \cite{BK}. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to $+$-associatedness of functions (as proved by Boczek and Kaluszka), but also to their $\star$-associatedness with $\star$ being an arbitrary strictly monotone and right-continuous binary operation. The second open problem deals with an existence of a pair of binary operations for which the generalized upper and lower Sugeno integrals coincide. Using a fairly elementary observation we show that there are many such operations, for instance binary operations generated by infima and suprema preserving functions.

Keywords:

binary operation, Sugeno integral, comonotone functions, $\star $-associatedness

Classification:

26A48, 28E10

paper.pdf

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