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In this paper, we discuss the hypothesis that an ordered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula>-semigroup can be constructed on the <i>M</i>-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula>-semigroups and ordered semigroups, we examined <i>M</i>-tri-ideals from a purely algebraic standpoint. We also present the form of the <i>M</i>-tri-ideal generator. We investigated the <i>M</i>-left(right)-tri-ideal using the ordered <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Γ</mo></semantics></math></inline-formula>-semigroup. In order to obtain their properties, we used <i>M</i>-left(right)-tri-basis. It was possible to generate a <i>M</i>-left(right)-tri-basis from elements and their subsets. Throughout this paper, we will present an interesting example of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>⪯</mo><mrow><mi>m</mi><mi>l</mi><mi>t</mi></mrow></msub><mrow><mo>(</mo><msub><mo>⪯</mo><mrow><mi>m</mi><mi>r</mi><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is not a partial order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula>. Additionally, we introduce the notion of quasi-order. As an example, we demonstrate the relationship between <i>M</i>-left(right)-tri-basis and partial order.