Partial Orders On U- Superabundant Semigroups

In the theory of semigroups, the regular semigroups have standed at a dominant status. And the generalized regular semigroups also play a vital role. As a result, the regular semigroups and generalized regular semigroups have been widely studied by a large number of scholars, both at home and abroad. Among them, the study on U- superabundant semigroups has been an important topic in the field of generalized regular semigroups.Suppose that S is a semigroup, and SE)( is the set of idempotents of S. Fixed a nonempty subset U in SE)( is called as a projections of S. A semigroup S, is called as a U- semiabundant semigroup, if each L-class and each R- class of which contains the element of U. A semigroup S, is called as a U- abundant semigroup, if it satisfies the compatible condition,that is, L is right compatible and R is left compatible.A U- abundant semigroup, denoted by(S, U), is called as a U- superabundant semigroup, if each H- class of which contains the element of U. Obviously, both the complete regular semigroups and the superabundant semigroups belong to the U- superabundant semigroups. And the U- superabundant semigroup is a common generalization of the complete regular semigroups and the superabundant semigroups.In this paper, with the help of the idempotent in the H- class and two preorders rω and lω on the idempotents set, we first defined the three relationships,l r≤ ≤ and ≤ on the U- abundant semigroup.Secondly, it is proved that the relationships,l r≤ ≤ and ≤ are partial orders on the U- abundant semigroup. And we discuss that the properties of U- abundant semigroup by the three partial orders.At last, we show that both l≤ is left compatible and right compatible with regard to the multiplication on the U- abundant semigroup(S, U), if and only if(S, U) is a local generalized clifford semigroup.