Exploring some saturated and closed varieties of semigroups

In a concrete category, a surjective morphism is always an epimorphism but the converse is not always true in the category of semigroups. However, the converse holds for saturated varieties of semigroups. There has been considerable effort invested in identifying saturated varieties of semigroups; however, achieving a complete characterization of these varieties still eludes researchers, posing an ongoing challenge. In this line of inquiry, we demonstrate that a semigroup variety conforming to the identity x1x2x3y = xi(1)xi(2)xi(3)y(2), where i is any nontrivial permutation on the set {1, 2, 3} is indeed saturated. Additionally, we illustrate that the varieties defined by the identities axy = ayxa and axy = ayax are each closed.