Some Studies For The Translational Hulls On Generalized Regular Semigroups

The translational hull of a semigroup plays an important role in the theory of ideal extensions of semigroups. The study of the translational hulls of regular semigroups is always one of the important topics in semigroups. In 1961, L.M.Gluskin studied firstly the translational hull of a semigroup in 《Ideals of semigroups》 , depicted the structure of a weakly reductive semigroup, and inaugurated the translational hull theory. In the next forty years, M.Petrich did many significant work. Nowadays, the translational hulls of generalized regular semigroups, which is the generalization of regular semigroups, have attracted more and more attention.By defining the left and right translation maps, the structure of the translational hulls in the generalized regular semigroups is provided in this paper. Firstly, by the definition of an E- semiadequate semigroup and Green relations, the translational hull of an Ehresmann semigroup is studied, and it is proved that the translation hull of an Ehresmann semigroup is still an Ehresmann semigroup. Secondly, an A-wrpp semigroup is defined, and it is proved that the translational hull of the A-wrpp semigroup is still an A-wrpp semigroup. Lastly, the structure of the translational hull of an adequate semigroup is investigated and it is proved that the semilattice of the translational hull is isomorphic to the translational hull of a semilattice, that is, EΩ (?)Ω(Es).