2019
Том 71
№ 11

# Velichko I. G.

Articles: 2
Brief Communications (Russian)

### Example of a function of two variables that cannot be an $R$-function

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 270–274

We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function $x_1 x_2 − 1$ possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an $R$-mapping for a proper choice of a surjection.

Brief Communications (Russian)

### Classification of topologies on finite sets using graphs

Ukr. Mat. Zh. - 2008. - 60, № 7. - pp. 992–996

With the use of digraphs, topologies on finite sets are studied. On this basis, a new classification of such topologies is proposed. Some properties of T0-topologies on finite sets are proved. In particular, it is proved that, in T0-topologies, there exist open sets containing arbitrary number of elements that does not exceed the cardinality of the set itself.