Lattice of cubes

A new type of lattices, lattice of cubes, is defined and described. It is proved that the number of all subcubes of the cube of the dimension m is 3 ^m . It is shown that the set of such subcubes with an appropriate choice of the operations of union and intersection is a lattice, called the lattice of cubes. An algorithm of constructing this lattice is proposed, and the problem of minimization and maximization of supermodular functions is considered on it. Particular examples of such functions are given. Optimizations algorithms, as well as the possibility of setting and solving a new class of problems on the cubes lattices are discussed.