Representation of right zero semigroups and their semilattices by a transformation semigroup

Background. As is known, an arbitrary semigroup can be represented by a semigroup of transformations that are right shifts either in this semigroup itself or in the extended semigroup obtained from the original one by adding an outer unit. The problems of finding all possible representations of a given semigroup, as well as applying the obtained theoretical results are being relevant. We studied semigroups of idempotents S, which are a semilattice of a finite number of semigroups of right zeros. Previously, one of the authors studied the exact matrix representations of such semigroups, while also considering the corresponding transformation semigroups of its subsemigroup, which is a minimal ideal in S. The purpose of this work is to apply theoretical research in the field of representation theory of semigroups to estimate the number of right zero semigroups and their semilattices and obtain a new semigroup invariant. Materials and methods. The work uses general methods of analysis and synthesis. Special methods of the theory of semigroups are also used: the method of establishing a homomorphism, the method of decomposing a semigroup of idempotents into a semilattice of rectangular semigroups, the method of partitions. To obtain quantitative estimates, a computer simulation method is used. Results. The study considers an idempotent semigroup S, which is a semilattice of a finite number of right zero semigroups. An exact representation of the semigroup S by a transformation semigroup is presented. A numerical sequence is found that is a complete invariant of the semigroup S. Based on the representation obtained, a general estimate is given for the number of nonisomorphic semilattices of two semigroups of right zeros. For semigroups of order from 2 to 20, a table of program results is given. An example demonstrates their consistency with theoretical conclusions. Conclusions. The results obtained can be used to establish the isomorphism of semigroups, since they allow cutting out inappropriate variants. We have to note that the asymmetry of the obtained results and the fact that for $k>m$ $N_{km}$ is always bigger than for $N_{mk}$. This means that, for a semigroup of a fixed order, if the number of elements in the first semigroup, which is a minimal ideal, is greater than in the second, then the number of nonisomorphic semilattices of the type under consideration is greater than in the opposite case.