THE CHROMATIC NUMBER OF THE FUNCTION

Abstract

The construction of the mathematical objects of the group structure on the set which is under the study and the use of the properties of this structure is one of the effective methods of the study. The concept of homomorphism is one of the basic concepts of group theory. This concept is very useful under the study of the properties of the groups. Homomorphism is the mapping from one group to another which preserves the group operation. An analogue of the concept of homomorphism in the case when an arbitrary everywhere defined mapping f : X n → X is given instead of the group operation has been constructed by the authors. The case when n = 2 and X ⊂ R have been studied in details in the article. The concept of the chromatic number of this mapping and the examples of its calculation have been given. The examples of the chromatic numbers of the certain groups have been given with the necessary explanations. The concept of the chromatic number of the real numerical function has been introduced. It has been shown that this concept is closely linked to the concept of V.L. Rvachev R - function. It has been shown, using the known results, that the functions with the infinite chromatic numbers exist. The examples of the chromatic numbers for the certain functions have been given with the necessary explanations. The main result of this article is the proof of the fact that the linear function of two real variables f (x, y) =αx + βy +γ ,αβ ≠ 0 has no finite chromatic number. The similar result has been proved for the function g(x, y) = x2 − y2 of two real variables. Thus, the set R can not be colored into the finite number of the colors in such a way that the color of the value of the function αx + βy +γ , where αβ ≠ 0 is uniquely determined by the colors of its arguments. The same fact is true for the function x2 − y2 and ax x x b n ... + 1 2 , where n >1, ab ≠ 0 . The obtained result can be formulated in terms of R - function as follows: the functions f (x, y) and g(x, y) (as well as the function ax x x b n ... + 1 2 under n >1, ab ≠ 0 ) can not be R - function at any choice of the accompanying functions of multiple-valued logic. Thus, the concepts of the chromatic class of the function and the chromatic number of the function have been introduced in the given article. The relation between the obtained concepts and group theory has been found. It has been demonstrated that the concept of the chromatic number of the function on the certain set is closely linked to the concept of V.L. Rvachev R - function. It has been pointed out that the fact that the chromatic numbers and the chromatic classes coincide for the isomorphic groups can be used under proving of the nonisomorphy of the groups.