In this paper, some properties of the ( k ∗ , k ) -lower part of ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals are obtained. Then, we characterize regular ordered semigroups in terms of its ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy generalized bi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left ideals and ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy right ideals, and an equivalent condition for ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left (resp. right) ideals is obtained. Finally, the existence theorems for an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal as well as for the minimality of an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal of an ordered semigroup are provided.

In this paper, after introducing the notion of relative bi-ideals and relative quasi ideals in ordered semigroups, some important properties of these bi-ideals and quasi ideals are studied. Then relatively prime and relatively weakly semiprime bi-ideals are defined and some vital results have been proved. We also define relative regularity and relative intraregularity of an ordered semigroup and prove some results based on the connection among intra-regularity of an ordered semigroup, relative quasi and relative bi-ideals of that ordered semigroup. Finally some important results connecting relative regularity, relatively prime bi-ideals and relatively weakly semiprime bi-ideals of an ordered semigroup have also been obtained.

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