A Class of Quasi-Adequate Semigroups

Let S be a quasi-adequate semigroup and the relation δ on S be defined by the rule: for a, b ∈ S, aδb if and only if E(a+)aE(a*) = E(b+)bE(b*) for some a+, a*, b+, b*. In this paper, we first construct an algebraic class called a QT-system and then prove that an abundant semigroup S is quasi-adequate on which δ is a congruence if and only if it is isomorphic to a QT-system.