Supermodular programming on finite lattices

There is a discription of the problems of minimization of supermodular functions on the different types of lattices: Boolean lattices, lattices with relative supplements (division lattices, lattices of vector subspaces of finite-dimensional vector space, geometrical lattices), lattices equal to Cartesian product of chains. The previously obtained theoretical results, on the basis of which the problems of minimization of supermodular functions on these lattices have been solved, are shown. A new type of lattices, lattice of Cubes, is defined and described. The problems of minimization and maximization of supermodular functions are considered on it. Particular examples of such functions are given. Optimization algorithms and the possibilities of setting and solving a new class of problems on the lattices of Cubes are discussed.