# Article

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Keywords:
comonotone functions; binary operation; \$\star \$-associatedness; Sugeno integral
Summary:
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.
References:
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