Some Studies Of Congruences On The Weakly Type ?-semigroups

A semigroup S is called a semi abundant semigroup,if every L-class and every R-class of S contain an idempotent.A semigroup S is called a U-semiabundant semigroup,if every LU-class and every RU-class of U contain an idempotent.The semiabundant semigroups and the U-semiabundant semigroups are extension of the regular semigroups.With the development of semigroup algebra theory,the research of these semigroups have attracted many scholars attention.By means of the generalized Green relation,the research mainly focus on a class of quasi-semiadequate semigroups,which is called a weakly type ?-semigroup.A quasi-semiadequate semigroup S is called a weakly type ?-semigroup,if each R-class of S contains a unique idempotent,? is a congruence and R is a left congruence,after the notion of quasi-spinded product of semigroups is introduced,a general structure of such a semigroup is obtained.It is proved that a semigroup S is a weakly type ?-semigroup if and only if S is a quasi-spinded product of a semiadequate semigroup T and a left regular band I.Next,based on the structure theorem of weakly type ?-semigroup,congruences on the weakly type ?-semigroups are studied by using the concept of good congruence pairs on the weakly type ?-semigroup,we establish a one-to-one correspondence relationship between the good congruence pair of S and its good congruence.At the end of this paper,U-weakly type ? semigroup is defined and some properties of U-weakly type ?semigroup is discussed,and the general structure of U-weakly type ? semigroup is established.