Abstract
In this paper, we introduce the relations I-m(gamma), I-gamma(n), B-gamma(m)n, (gamma)Q(m)(n) and H-gamma(m)n on an le-Gamma-semigroup and prove that the inclusions B-m(n) subset of Q(m)(n) subset of H-m(n) hold on le-Gamma-semigroups and provide some sufficient conditions under which equalities hold in these relations. Further, we show that the (m, 0, gamma)-regularity [(0, n, gamma)-regularity] of an element of any le-Gamma-semigroup induces the (m, 0, gamma)-regularity [(0, n, gamma)-regularity] of the whole (gamma)Q(m)(n)-class and H-gamma(m)n-class containing that element. We further show that the (m, n, gamma)-right weakly regularity of an element induces the (m, n, gamma)-right weakly regularity of the whole B-gamma(m)n-class, (gamma)Q(m)(n)-class and H-gamma(m)n-class containing that element. Finally, we introduce the notion of a strong (m, n, alpha, beta)-quasi-ideal element in an le-Gamma-semigroup and provide some sufficient conditions under which each (m, n, alpha, beta)-quasi-ideal element becomes a strong (m, n, alpha, beta)-quasi-ideal element.