$\sigma$-Extensions of discrete multivalued groups

The aim of this paper is to discover new classes of discrete multivalued groups and to find criteria for recognition of coset groups. It is shown in [11] that the set of discrete coset groups with one generator is the union of categories. These categories are indexed by the set of special symmetric graphs. Bearing in mind our aim, we define $\sigma$-extensions of discrete multivalued groups. Our main result is as follows: the construction of $\sigma$-extensions of discrete multivalued groups is equivariant with respect to functors of these categories.