Componentwise decomposition of some lattice-valued fuzzy integrals

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set, the top element of the lattice to the entire universe and satisfy the property of monotonicity. If the lattice is complete then a lattice-valued fuzzy integral of Sugeno type, with similar properties such as the Sugeno integral in its original form, can be introduced in a natural way. The main result of the paper is a componentwise decomposition theorem of an LL-valued fuzzy integral to its L-valued fuzzy integrals components, where L   is a complete lattice with negation and L={(α,β);α,β∈L,α⩽β¯} is organized as a complete lattice too. This result is useful to obtain the properties of LL-valued fuzzy integrals from the properties of L-valued fuzzy integrals and to calculate in a simple way the values of some integrals from the values of the components. The important case L = [0, 1], when LL becomes the lattice of the intuitionistic fuzzy values is distinctly discussed. An idea of application to synthetic evaluation of objects is also suggested.