on dominions and determination of closedvarieties of semigroups

It is known that all subvarieties of variety of all semi groups 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 semi groups. 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 com mutative semigroups to some generalized classes of commutative semigroups by showing that dominions of such semigroups belong to the same class.