Algebra of Finite-Valued Functions: Classification of Functions and Subalgebras, Essential and Fictitious Subalgebras

The classification of subalgebras of every algebra of finite-valued functions is constructed. Classes of this classifications do not intersect. Each class contains subalgebras with the same number of functions in minimal basis. Class with ordinal number 0 contains subalgebras that have no basis. The class with finite ordinal number n contains subalgebras whose minimal basis has n functions. The set of subalgebras of this class are countable. There is a class with infinite ordinal number. Subalgebras of this class have a minimal basis with infinite number of functions. The set of these subalgebras is continual. Only the class with ordinal number 1 is essential, all other classes are fictitious, since they are useless to classify functions. But classification of functions is the main problem of the algebra of finite-valued functions. A class of this classification is a set of functions extracted from one-member bases of a subalgebra. Each function generates by superpositions some subalgebra, and only this subalgebra. So, this function belongs to only one class. All these classes of functions belong to the class 1 of subalgebras. All subalgebras from classes with the other ordinal numbers are useless to classify functions. The set of these fictitious subalgebras is continual, the set of essential subalgebras are countable. The top level of the classification of functions contains the algebra of finite-valued functions. Next level contains maximal subalgebras. According to I.G. Rosenberg, there are 6 sets of maximal subalgebras. I.G. Rosenberg was wrong to state the set of his quasilinear functions be maximal. Only Yablonsky’s set of quasilinear functions is maximal. The sixth Rosenberg’s set also turns to be wrong. This right set was built by A.I. Maltsev. But from 6 sets only 3 sets contain essential subalgebras. And all maximal essential subalgebras containing 3-valued 2-place functions are built.