In this work, we introduce the concept of \((\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\mathfrak {q}_{\textrm{k}})})\) -fuzzy quasi-ideals in ordered semigroups. We provide several equivalent conditions of these fuzzy quasi-ideals and demonstrate that the intersection of any collection of these fuzzy quasi-ideals is of the same type, but their union is not. We also discuss how these novel fuzzy quasi-ideals connect to previously studying fuzzy quasi-ideal notions in the literature. Additionally, we provide characterizations of \((\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\mathfrak {q}_{\textrm{k}})})\) -fuzzy quasi-ideals in terms of \((\textrm{k}^*,\textrm{k})\) -upper part of fuzzy sets. We further introduce the notion of completely semiprime \((\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\mathfrak {q}_{\textrm{k}})})\) -fuzzy quasi-ideals and establish several equivalent conditions for them in terms of \((\textrm{k}^*,\textrm{k})\) -upper parts of fuzzy sets and ordered fuzzy points. Finally, we characterize the left and right regularities of ordered semigroups by using completely semiprime \((\overline{\in },\overline{\in }\vee \overline{(\textrm{k}^*,\mathfrak {q}_{\textrm{k}})})\) -fuzzy quasi-ideals.
