Exploring some saturated and closed varieties of semigroups
Shabir Ahmad Ahanger, Kulsooma Rashid, Manan Bashir Bhat · Asian-European Journal of Mathematics · 2025
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 [Formula: see text] where [Formula: see text] is any nontrivial permutation on the set {1, 2, 3} is indeed saturated. Additionally, we illustrate that the varieties defined by the identities [Formula: see text] and [Formula: see text] are each closed.