Geodesic
On $\star $- associated comonotone functions
Kybernetika, Tome 54 (2018) no. 2, pp. 268-278

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We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. 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.
DOI : 10.14736/kyb-2018-2-0268
Classification : 26A48, 28E10
Keywords: comonotone functions; binary operation; $\star $-associatedness; Sugeno integral 
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Hutník, Ondrej; Pócs, Jozef. On $\star $- associated comonotone functions. Kybernetika, Tome 54 (2018) no. 2, pp. 268-278. doi: 10.14736/kyb-2018-2-0268

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