Ordered Quasi(BI)-Γ-Ideals in Ordered Γ-Semirings

<jats:p>In this paper, we have defined ordered quasi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals and ordered bi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals in ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-semirings by defining the relation “<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mo>≤</mml:mo></mml:mrow></mml:math>” in ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math> semiring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>a</mml:mi><mml:mo>≤</mml:mo><mml:mi>b</mml:mi></mml:math> if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>b</mml:mi></mml:math> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi><mml:mo>.</mml:mo></mml:math> By using this relation we have shown that ordered quasi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals and ordered bi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals in ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-semirings are generalization of quasi-ideals and bi-ideals in ordered semirings. Properties of many types of ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals including (semi)prime, (strongly) irreducible, and maximal ordered quasi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals and ordered bi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-ideals in ordered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math>-semirings <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math> have been studied.</jats:p>