A Novel Method for Generating the <i>M</i>-Tri-Basis of an Ordered <i>Γ</i>-Semigroup

In this paper, we discuss the hypothesis that an ordered Γ-semigroup can be constructed on the M-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using Γ-semigroups and ordered semigroups, we examined M-tri-ideals from a purely algebraic standpoint. We also present the form of the M-tri-ideal generator. We investigated the M-left(right)-tri-ideal using the ordered Γ-semigroup. In order to obtain their properties, we used M-left(right)-tri-basis. It was possible to generate a M-left(right)-tri-basis from elements and their subsets. Throughout this paper, we will present an interesting example of order ⪯mlt(⪯mrt), which is not a partial order of S. Additionally, we introduce the notion of quasi-order. As an example, we demonstrate the relationship between M-left(right)-tri-basis and partial order