Lattice of cubes (Q733913)

The authors find the number of subcubes of the cube in arbitrary dimension and show 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. Optimization algorithms, as well as the possibility of setting and solving a new class of problems on the cube lattices are discussed.