On Relation $\mathcal Q_\gamma$ in $le$-$\gamma$-Semigroups

Petraq Petro, Kostaq Hila, Jani Dine

In this paper we introduce and study the relation $\mathcal Q$ in $le$-$\Gamma$-semigroups. This relation in general turns out to have better properties than the relation $\mathcal H_\gamma$ studied in [10]. We give several properties that hold in every $\mathcal Q_\gamma$-class of an $le$-$\Gamma$-semigroup and especially in every $\mathcal Q_\gamma$-class satisfying the Green's condition. In particular, the $\gamma$-regularity and $\gamma$-intra-regularity of a $\mathcal Q_\gamma$-class is studied. We also consider a case a $\mathcal Q_\gamma$-class of an $le$-$\Gamma$-semigroup $M$ forms a subsemigroup of $M_\gamma=(M,\circ)$.