Generalized Bruck-Reilly Semigroups, Bisimple And Simple Inverse ω~2-Semigroups

The set N of all nonnegative integers forms a chain under the reverse of usual number order called theω—chain. The set E—N x N un-der the lexicographic order is called theω~2—chain. In this paper, the Munn's semigroup T_E is characterized by so-called double bicyclic semigroup Be- A new kind semigroups called generalized Bruck-Reilly semigroups GBR(T;β,γ; u, e) is defined which is a generalization of the so-called bisimple inverseω~∞-semigroup GBR(T;β,γ; u), due to Shang Yu and Wang Limin. Some basic properties of this new kind of semigroups, such as Green's relations, regularity,π—inverse and simple properties, are discussed. In the main parts of this paper, the homomor-phisms and congruences of a bisimple inverseω~2-semigroup S are investigated. It is proved that groups, bisimple inverseω—semigroups and bisimple inverseω~2-semigroups are only homomorphic images of S and that the congruences on S are also group-, bisimple inverseω-semigroup and idempotent-separating con-gruences only. An isomorphism theorem for bisimple inverseω~2-semigroups is in passing obtained. Moreover two special simple inverseω~2-semigroups is also investigated. The results obtained here can be viewed preliminaries for further studies of simple inverseω~2-semigroups.