Some Studies On U-superabundant Semigroups

The study of the class of regular semigroups and its subclasses have been a dominantdirection in the theory of semigroups. As the development of semigroup theory, many authorsfocus on the classes of generalized regular semigroups . The study of U-abundant semigroupshas become an important topic in the study of generalized regular semigroups.Firstly, we de?ne and study U-cryptoabundant semigroups. A U-superabundant semi-group S is called a U-cryptoabundant semigroup, if the generalized Green's relation HU is acongruence on S. U-cryptoabundant semigroups are a particular class of U-superabundantsemigroups. U-cryptoabundant semigroups can be seen as a kind of generalization of cryp-togroups in the class of U-abundant semigroups.Secondly, we study some important properties of DU -majorization in U-superabundantsemigroups. And some equivalent statements of DU-majorization are given in this paper.In this paper, the de?nition of normal U-cryptoabundant semigroups is given by meansof regular band. A U-cryptoabundant semigroup is called a normal U-cryptoabundant semi-group if S/HU is a normal band. We proved that a U-superabundant semigroup S is a normalU-cryptoabundant semigroups, if and only if, for any projection e, eSe is a generalized Clif-ford semigroup; if and only if, S satis?es DU -majorization and HU is a congruence; if andonly if, S is a strong semilattice of completely J -simple semigroups and HU is a congruence.As the corollaries, we obtain some equivalent statements of normal cryptogroups andnormal cryptoabundant semigroups in the classes of completely regular semigroups and su-perabundant semigroups, respectively. Finally, quasi-superabundant semigroups are de?ned and semilattice decomposition ofsuch kind of semigroups is given in this paper.