  Despite the abundant literature on the subject appeared in the last few
years, the lattice Boltzmann method (LBM) is probably the one for which a
complete understanding is not yet available. As an example, an unsolved
theoretical issue is related to the construction of a discrete kinetic theory
which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic
(here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper
aims at investigating discrete inverse kinetic theories (IKT) for
incompressible fluids. We intend to show that the discrete IKT can be defined
in such a way to satisfy, in particular, the requirement of
\emph{completeness}, i.e., {\it all} fluid fields are expressed as moments of
the kinetic distribution function and {\it all} hydrodynamic equations can be
identified with suitable moment equations of an appropriate inverse kinetic
equation IKE.
