  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.
