Congruence Relations On Double Cycle Semigroups

The bicyclic semigroups is a class of special inverse semigroups. In this paper using the congruence relations on the bicyclic semigroups, we give the detailed description about the structure and properties of the bicyclic semigroups. The main results are as follows:1. By researching a class of congruences on the bicyclic semigroups, we studied the congruence class which contains all idempotent elements, proved that such congruence class is an inverse semigroup, and then gives the congruence class which contains all idempotent elements on the bicyclic semigroups is a regular semigroup.2. This section is devoted to study a class of congruences (ρd)d∈N on a bicyclic semigroup. It is shown that the quotient semigroup <N×N,·>/∩d∈N ρd forms a group, and such group is characterized. To investigate the homomor-phism between bicyclic semigroups, a homomorphism from bicyclic semigroup to the integer addition semigroup is given, and it is proven that the kernel of such homomorphism is the least group congruence on bicyclic semigroup.3. This paper is devoted to study congruences on a bicyclic semigroup. It is shown that there is an unique bijection between a class of congruences ρd(d∈ N) on it and its inverse subsemigroups. Also, the lattice of such congruences and the lattice of natural numbers under some partial order are isomorphic, and then give some important result relating to the green’s equivalence relation.