In this article, we try to obtain a more general form than ( ∈ , ∈ ∨ q ) -intuitionistic fuzzy bi-ideals in ordered semigroups. The notion of an ( ∈ , ∈ ∨ q k ) -intuitionistic fuzzy bi-ideal is introduced, and several properties are investigated. Characterizations of an ( ∈ , ∈ ∨ q k ) -intuitionistic fuzzy bi-ideal are established. A condition for an ( ∈ , ∈ ∨ q k ) -intuitionistic fuzzy bi-ideal to be an intuitionistic fuzzy bi-ideal is provided. It is shown that every ( ∈ , ∈ ) -intuitionistic fuzzy bi-ideal is an ( ∈ , ∈ ∨ q ) -intuitionistic fuzzy bi-ideal, and every (∈,∈∨q)-intuitionistic fuzzy bi-ideal is an (∈,∈∨qk)-intuitionistic fuzzy bi-ideal but the converse is not true. The important achievement of the study with an ( ∈ , ∈ ∨ q k ) -intuitionistic fuzzy bi-ideal is that the notion of an (∈,∈∨q)-intuitionistic fuzzy bi-ideal is a special case of an ( ∈ , ∈ ∨ q k ) -intuitionistic fuzzy bi-ideal, and, thus, several results in the paper (Jun et al. in Bi-ideals of ordered semigroups based on the intuitionistic fuzzy points (submitted)) are the corollaries of our results obtained in this paper.

In this paper, we introduce fuzzy bipolar soft semiprimality in the structure of ordered semigroups and investigate some properties of the concept. Moreover, ordered semigroups and their some classes are characterized by means of fuzzy bipolar soft semiprimality. Finally, the Cartesian product of fuzzy bipolar soft semiprime (resp., prime) ideals over ordered semigroups is examined. Some of the ideas are supported by apt examples.

A new trend of using fuzzy algebraic structures in various applied sciences is becoming a central focus due to the accuracy and nondecoding nature. The aim of the present paper is to develop a new type of fuzzy subsystem of an ordered semigroup S. This new type of fuzzy subsystem will overcome the difficulties faced in fuzzy ideal theory of an ordered semigroup up to some extent. More precisely, we introduce ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals of S. These concepts are elaborated through appropriate examples. Further, we are bridging ordinary ideals and ∈,∈∨qkδ-fuzzy ideals of an ordered semigroup S through level subset and characteristic function. Finally, we characterize regular ordered semigroups in terms of ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals.

Several applied fields dealing with decision-making process may not be successfully modeled by ordinary fuzzy sets. In such a situation, the interval-valued fuzzy set theory is more applicable than the fuzzy set theory. Using a new approach of “quasi-coincident with relation”, which is a central focused idea for several researchers, we introduced the more general form of the notion of ( α , β )-fuzzy interior ideal. This new concept is called interval-valued ( ∈ , ∈ ∨ q k ˜ ) $( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$ -fuzzy interior ideal of ordered semigroup. As an attempt to investigate the relationships between ordered semigroups and fuzzy ordered semigroups, it is proved that in regular ordered semigroups, the interval-valued ( ∈ , ∈ ∨ q k ˜ ) $( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$ -fuzzy ideals and interval-valued ( ∈ , ∈ ∨ q k ˜ ) $( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$ -fuzzy interior ideals coincide. It is also shown that the intersection of non-empty class of interval-valued ( ∈ , ∈ ∨ q k ˜ ) $( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$ -fuzzy interior ideals of an ordered semigroup is also an interval-valued ( ∈ , ∈ ∨ q k ˜ ) $( \in ,{\rm{ }} \in \; \vee \;{{\rm{q}}_{\tilde k}})$ -fuzzy interior ideal.