Some Applications on the Second Order Logic Language in the Universal Algebra

By consideration of the problematic of the universal algebra as some questions which are connected with some collection of functions (signiches functions), which are defined on some set (the basic set of algebra) take place the natural interest too some functions which can be defined in this set by the help, in some sense, from the signiches functions. The only restriction by this is the natural for the universal algebra (which study the algebras to the nearest by isomorphism) is the requirement for this functions to be permutable with the automorphisms of starting algebra. The functions which are defined on the algebra with the help of some logical language (the language of the first order logic, or the language of the logics with the infinite formulas, or the language of the second order logic and so other) are such. In the work is considered the questions which are connected with the functions and elements which are defined on the countable algebras of finite signatures in the language of the second order logic. It is known the Marek-theorem: by the preposition of set-theoretical axsiom of constructivity the theory of countable universal algebras of finite signatures in the language of second order logic are categorical. From this theorem it is proved that the functions on the basic set of this algebras which commutes with automorphisms of this algebra can be defined on this algebra in some natural extension of the second order logic language. As some corollary of this we given some description of endomorphisms (automorphisms) of countable universal algebras of finite signatures which commutes with oll automorphisms of this algebras. Another corollary given some description of elements from Galya-closuring of subalgebras of countable universal algebras of finite signatures.