Some Studies On The Translational Hull Of Some Generalized Regular Semigroups

The study of the translational hull of semigroups is an important topic in semigroup theory and plays a basic role in the theory of ideal extensions of semigroup. In 1973, Alut studied the translational hull of inverse semigroups. In 1985, Fountain and Lawson further investigated the translational hull of adequate semigroups. In addition, the translational hulls of some kinds of semigroups were further studied and many results were obtained. Such as the translational hull of weekly reductive semigroups, the translational hulls of completely 0-simple semigroups and Rees inverse monoids have been studied and depicted. Since the generalized regular semigroups are the generalization of regular semigroups, the translational hulls of this kind of semigroups get more and more attention.In this paper, we first recall the concept of the left and right translation maps and the translational hull of semigroups. By using some generalized regular semigroup and (?) relation, the translational hulls of a strongly right Ehresmann semigroup and a strongly right U-ample semigroup are studied, We now call a semigroup (S, U) a U-rpp semigroup if every (?)-class of (S, U) contains a projection of (S, U). A U-rpp semigroup (S, U) is called a right Ehresmann semigroup if the projections of (S, U) commute and (?) on (S, U) is a right congruence. A U-rpp semigroup (S, U) is called strongly U-rpp semigroup if there is a unique projection e such thatα(?)e andα= eα, for anyα∈(S,U). Thus, we naturally call a right Ehresmann semigroup (S, U) is a strongly Ehresmann semigroup if (S, U) is a strongly U-rpp semigroup. A strongly right U-ample semigroup (S, U) is a strongly right Ehresmann semigroup in which eS∩αS= eαS for every element a and every projection e ofThis research is supported by National Natural Science Foundation of China(Grant No:10971160) and the NSF grant of Shaanxi Province(SJ08A06). S. And in this paper it is proved that:the translational hull of a strongly right Ehresmann semigroup (a strongly right U-ample semigroup) is itself strongly right Ehresmann (strongly right U-ample).Secondly, considering that U-ample semigroup is a special kind of Ehresmann semigroup, the paper probes into the study of the translational hull of a U-ample semigroup and obtain the similar result.Finally, we obtain the generalized Green relation on the translational hull of U-ample semigroup in this paper.