Given a possibilistic distribution on a nonempty space Ω with possibility degrees in a chained complete lattice, the lattice-valued entropy function for such distribution is defined as the expected value (in the sense of Sugeno possibilistic integral) of the lattice-valued function ascribing to each ωeΩ the possibilistic measure of its complement Ω-{ω}.
However, such an entropy function seems to be little sensitive or flexible in the sense that it ascribes the same and supremum value to a rather wide class of different lattice-valued possibilistic distributions so that the choice of the most adequate, in a sense, distribution (for the purposes of decision making under uncertainty) is rather limited. In this paper, we propose and analyze a refined version of this entropy, which splits the wide class of possibilistic distributions mentioned above into a rich spectre of narrower classes of distributions to which different but mutually comparable values of the refined entropy function are ascribed.