On Dominions and Determination of Closed Varieties of Semigroups

Document Type : Research Paper


Department of mathematics, Aligarh Muslim University, Aligarh-202002, India.


  It is known that all subvarieties of variety of all semigroups are not absolutely closed. So, it is a natural question to find out those subvarieties of variety of all semigroups that are closed in itself or close in larger subvarieties of variety of all semigroups. We have gone through this open problem and able to determine some closed varieties of semigroups defined by the identities $axy=yxax~[axy=xyxa]$ and $axy=yxxa$ by using Isbell's zigzag theorem as an essential tool. Further, we partially generalize a result of Isbell on semigroup dominions from the class of commutative semigroups to some generalized classes of commutative semigroups by showing that dominions of such semigroups belong to the same class.