Ordered quasi(BI)-\(\Gamma\)-ideals in ordered \(\Gamma\)-semirings (Q2337138)

Summary: In this paper, we have defined ordered quasi-\(\Gamma\)-ideals and ordered bi-\(\Gamma\)-ideals in ordered \(\Gamma\)-semirings by defining the relation ``\(\leq\)'' in ordered \(\Gamma\) semiring \(S\) as \(a \leq b\) if \(a + x = b\) for any \(a\), \(b\), \(x \in S\). By using this relation we have shown that ordered quasi-\(\Gamma\)-ideals and ordered bi-\(\Gamma\)-ideals in ordered \(\Gamma\)-semirings are generalization of quasi-ideals and bi-ideals in ordered semirings. Properties of many types of ordered \(\Gamma\)-ideals including (semi)prime, (strongly) irreducible, and maximal ordered quasi-\(\Gamma\)-ideals and ordered bi-\(\Gamma\)-ideals in ordered \(\Gamma\)-semirings \(S\) have been studied.