Generalized comonotonicity and new axiomatizations of Sugeno integrals on bounded distributive lattices

Two new generalizations of the relation of comonotonicity of lattice-valued vectors are introduced and discussed. These new relations coincide on distributive lattices and they share several properties with the comonotonicity for the real-valued vectors (which need not hold for $L$-valued vectors comonotonicity, in general). Based on these newly introduced generalized types of comonotonicity of $L$-valued vectors, several new axiomatizations of $L$-valued Sugeno integrals are introduced. One of them brings a substantial decrease of computational complexity when checking an aggregation function to be a Sugeno integral.