Saturated Varieties of Semigroups

The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially demonstrate that the variety of semigroups defined by the identity axy = ayxa is saturated. The next main result establishes that the variety of semigroups determined by the identity axy = ayax is saturated. Finally, we show that medial semigroups satisfying the identity xy = xy(n), where n >= 2, are also saturated. These results collectively lead to the conclusion that epis from these saturated varieties are onto. This paper thus offers substantial progress towards the comprehensive characterization of saturated varieties of semigroups.