## I. INTRODUCTION 

During the last decade the most impressive achievements in magnetism were related to fabrication, investigation, and application of artificial magnetic materials (see Ref. 1 for a recent review). The technologies of sputtering and lithography have progressed to the state where the manufacture of nanosize, periodic magnetic superlattices of different types is feasible. Among them two-dimensional lattices of sub-micron magnetic particles (so-called magnetic dots) attract much attention. These magnetic dots, in the form of circular or elliptic cylinders, or rectangular prisms, are made of soft magnetic materials such as Co and permalloy, 2–6 or highly anisotropic materials like Dy, 7 and FePt, 8 and the dot array lattice is usually designed to be quadratic or rectangular. In the array dots are separated from each other so that direct exchange interaction between dots is completely absent. Thus the dipolar interaction is the sole source of coupling between dots and determines the pattern of dot magnetic moment orientations that constitutes the physical properties of a dot array. Owing to the absence of exchange, magnetic dot arrays constitute promising material for high-density magnetic storage media. For this purposes, the dense arrays (with the period of order of 100-200 nm) of small enough magnetic dots with the magnetic moments perpendicular to the array plane are optimal, see Ref. 9. For small enough dots (the diameter < 100 nm) the magnetization inside of a dot is almost uniform, producing the total magnetic moment m$_{0}$ ≫ µ$_{B}$ , where µ$_{B}$ is Bohr magneton. Therefore, this can be a new kind of magnetic material with purely two-dimensional lattice structure and pure dipolar coupling between large enough magnetic moments. 

To describe the physical properties of magnets in general it is essential to take into account interactions having various origins and different energy scales. For adjacent spins the spin exchange interaction is almost invariably the strongest. If this interaction is ferromagnetic it causes uniform spin ordering over distances that usually substantially exceeds the atomic spacing. The dipole-dipole interaction, though as a rule is considerably weaker than the exchange for adjacent spins, but it extends to much longer range. The competition of these two interactions produces magnetic domain structure with long-range non-uniformity of magnetization for usual magnetic samples. 

Models of magnetic moments with the dipolar interaction have been theoretically studied for more than 60 years, and many physical properties, lacking in the spin-exchanged systems, 

