Basic properties of lattices of cubes, algorithms for their construction, and application capabilities in discrete optimization

The basic properties of a new type of lattices — a lattice of cubes — are described. It is shown that, with a suitable choice of union and intersection operations, the set of all subcubes of an $N$-cube forms a lattice, which is called a lattice of cubes. Algorithms for constructing such lattices are described, and the results produced by these algorithms in the case of lattices of various dimensions are illustrated. It is proved that a lattice of cubes is a lattice with supplements, which makes it possible to minimize and maximize supermodular functions on it. Examples of such functions are given. The possibility of applying previously developed efficient optimization algorithms to the formulation and solution of new classes of problems on lattices of cubes.