  A non-perturbative algebraic theory of lattice Boltzmann method is developed
based on a symmetry of a product. It involves three steps: (i) Derivation of
admissible lattices in one spatial dimension through a matching condition which
imposes restricted extension of higher-order Gaussian moments, (ii) Special
quasi-equilibrium distribution function found analytically in closed form on
the product-lattice in two and three spatial dimensions, and which proves
factorization of quasi-equilibrium moments, and (iii) Algebraic method of
pruning based on a one-into-one relation between groups of discrete velocities
and moments. Two routes of constructing lattice Boltzmann equilibria are
distinguished. Present theory includes previously known limiting and special
cases of lattices, and enables automated derivation of lattice Boltzmann models
from two-dimensional tables, by finding roots of one polynomial and solving a
few linear systems.
