Key words Soft set, soft ordered semigroup, soft ordered subsemigroup, idealistic soft ordered semigroup. MSC (2000) 03G25, 06D72, 06F05Molodtsov introduced 1999 the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to ordered semigroups. The notions of (trivial, whole) soft ordered semigroup, soft ordered subsemigroup, soft left (right) ideal, and left (right) idealistic soft ordered semigroup are introduced, and various related properties are investigated.

The intuitionistic fuzzy set, briefly as; IFS and its extension involving Pythagorean fuzzy set (PFS) and Fermatean fuzzy set (FFS), are all effective tools to express uncertain and incomplete cognitive information with membership, nonmembership, and hesitancy degrees. The FFS introduced by Senapati and Yager, carries out uncertain and imprecise information smartly in exercising decision-making than IFS and PFS. A generalized form of union and intersection on FFS can be formulated from a generalized T-norm and T-conorm. Hamacher operations such as Hamacher product and Hamacher sum, are good alternatives to product and sum. In this course of this article, we first device new operations on Fermatean fuzzy information by employing Hamacher T-conorm and T-norm and discuss basic operations. Induced by the Hamacher operations and FFS, we propose Fermatean fuzzy Hamacher arithmetic and also geometric aggregation operators. In the first section, we introduce the concepts of a Fermatean fuzzy Hamacher weighted average operator, a Fermatean fuzzy Hamacher ordered weighted average operator, and a Fermatean fuzzy Hamacher hybrid weighted operator and discuss their basic properties in detail. In the second part, we develop Fermatean fuzzy Hamacher weighted geometric operator, Fermatean fuzzy Hamacher ordered 2 2 , which does not carry out the boundary condition of a PFS. However, it is clear to see (0.8) + 3 (0.7) = 0.512 + 0.343 = 0.855 < 1 3 , which is an appropriate reason for the introduction of a HADI ET AL.| 3465 new division of fuzzy set, called Fermatean fuzzy set (FFS). It is also important to mention that the class of this type of fuzzy sets has more ability to capture the uncertainties as compared with IFSs and PFSs, and are qualified to handle higher degree of vagueness. MADM has been extensively used in many area of sciences, for example, Xu and Xia 17 ; and Xu and Chen 18 introduced intuitionistic fuzzy weighted averaging (IFWA) operator and intuitionistic fuzzy ordered weighted averaging (IFOWA) operator.The fuzzy information aggregation operators are necessarily appealing in significant research areas and are given profound consideration among the scientific community.

and right weakly regular ordered semigroups Ideals, quasi-ideals and bi-ideals Fuzzy ideals, fuzzy quasi-ideals and fuzzy bi-ideals a b s t r a c tIn this paper we characterize different classes of ordered semigroups by the properties of their ideals, quasi-ideals and bi-ideals. We also characterize these classes by the properties of their fuzzy ideals, fuzzy quasi-ideals and fuzzy bi-ideals.

In this paper, we define fuzzy generalized bi-ideals in ordered semigroups and we characterize regular and left weakly regular ordered semigroups by the properties of their fuzzy generalized bi-ideals.Keywords: Fuzzy subsets; regular, left weakly regular ordered semigroups; left (resp. right, bi-and generalized bi-) ideals; fuzzy left (resp. right, bi-and generalized bi-) ideals in ordered semigroups.

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In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.

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