Biregular semigroups. II
The following problem is considered: when can the action of a cancellative semigroup on a set be extended to a simply transitive action of the universal group of on a larger set.
The article considers a problem from Trokhimenko paper [13] concerning the study of abstract properties of commutations of operations and their connection with the Menger and Mann superpositions. Namely, abstract characterizations of some classes of operation algebras, whose signature consists of arbitrary families of commutations of operations, Menger and Mann superpositions and their various connections are found. Some unsolved problems are given at the end of the article.
Commutative semigroups satisfying the equation and having only two -invariant congruences for an automorphism group are considered. Some classes of these semigroups are characterized and some other examples are constructed.