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In 2019, Öztürk et al. [Gamma semigroups on weak nearness approximation spaces, Journal of the International Mathematical Virtual Institute9 (2019) 53–72] defined the first algebraic structure on weak nearness approximation spaces which is the gamma semigroup. After this study, the view on the nearness of algebraic structures has completely changed. This view was first expressed in a previous study in 2021 [M. A. Öztürk, Nearness d -algebras, Journal of Algebraic Hyperstructures and Logical Algebras2 (2021) 73–84]. Our aim is to give the concept of modulo in the nearness group G as in the nearness group (Z , +) , where Z is the set of integers. Afterward, based on the concept of modulo in the nearness group G , the nearness cosets of the nearness group G are constructed and its basic properties are examined. Lastly, it is seen that Lagrange theorem is not valid for the nearness subgroups as compared to the usual subgroups. [ABSTRACT FROM AUTHOR]

In 2022, Öztürk defined the nearness equivalence classes and nearness cosets of the nearness groups (see [10]). Furthermore, he showed a nearness group G induced by a nearness subgroup H can not be written two different decompositions of G by separate left (right) cosets [10]. In other words, if H is a nearness subgroup of a nearness group G, then G may be shown as a union of different non-discrete left (right) cosets of H in G. Also, it is given that Lagrange's theorem does not work in the nearness subgroups as usual subgroups. Our purpose in this paper is to introduce normal nearness subgroups and quotient nearness groups. Besides which, it is given an example of a nearness group that has nearness subgroup but not normal nearness subgroup. Also, a criterion that satisfies the condition of being a normal subgroup is investigated. It is shown that the multiplication of the two right near-cosets (left near-cosets) is again a right near-coset (left near-coset) is valid in normal nearness subgroups under certain conditions. Moreover, examples of normal nearness subgroups and quotient nearness groups are given. [ABSTRACT FROM AUTHOR]

In this paper, we study some concepts of subgroups on weak nearness approximation spaces. We show some properties that are satisfied by usual subgroups are not satisfied in nearness subgroups. In addition to this, it is defined that nearness subgroup generated by a subset on weak nearness approximation spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the notion of ideals in gamma rings on weak nearness approximation spaces and explain some of the concepts and definitions. Also, an example is given related to the subject. The features described in this study will contribute greatly to the theoretical development of the gamma nearness rings theory. We defined that upper nearness ideal generated by a subset of the ring on weak nearness approximation spaces and introduce some properties of these ideals. In addition to this, prime nearness ideals and maximal nearness ideals of the gamma nearness rings are defined and some properties of these ideals are introduced. [ABSTRACT FROM AUTHOR]

In this paper, the notions of ideals in NBCK-algebras are presented and some properties of such ideals are analyzed. Essentially, our approach is to define NBCK-algebras of all cosets by considering an operation on the set of all cosets defined via ideals. Afterwards, our aim is to study ideal theorems on NBCK-algebras. [ABSTRACT FROM AUTHOR]

This paper is concerned with basic concepts and some results on regular Γ-nearness semigroup and ideals of a Γ-nearness semigroup. Also, it is given some properties about ideals of a regular Γ-nearness semigroup and an example about the subject. Furthermore, we study relations among ideals of a Γ-nearness semigroup. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce Nobusawa Γ -nearness hemiring theory and also to study some properties of ideals and prime (semiprime) ideals for Nobusawa Γ -nearness hemirings. [ABSTRACT FROM AUTHOR]

In this study, we investigate the problem of how to define Γ -nearness semiring in the sense of Nobusawa theory which extends the notion of a nearness rings, nearness semirings and Γ -rings [N. Nobusawa, On a generalization of the ring theory, Osaka Journal of Mathematics1 (1964) 81–89; M. A. Öztürk, Semirings on weak nearness approximation spaces, Annals of Fuzzy Mathematics and Informatics15(3) (2018) 227–241; M. A. Öztürk and E. İnan, Nearness rings, Annals of Fuzzy Mathematics and Informatics17(2) (2019) 115–132]. Also, we introduce some properties of approximations and their algebraic structures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of gamma nearness semiring theory which extends the notion of the nearness semiring [15] and study several properties. Also, the concept of Γ-ideals, Γ-prime ideals and Γ-semiprime ideals of nearness semiring theory is investigated and some properties of such ideals are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of a Γ -prime (respectively semiprime) ideals of the Γ -semigroup on 𝒪 − 𝒪 ´ , where (𝒪 , ℱ , ∼ B r , N r) and (𝒪 ´ , ℱ , ∼ B r , N r) are two different weak near approximation spaces. Several characterizations of them are furnished. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the concept of prime (semiprime) ideals of nearness semiring theory and to introduce some properties of such ideals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of how to define Γ -nearness ring in the sense of Nobusawa theory which extends the notion of a nearness ring and Γ -rings [N. Nobusawa, Osaka J. Math.1 (1964) 81–89; M. A. Öztürk and E. İnan, Annals of Fuzzy Mathematics and Informatics17(2) (2019) 115–131]. Also, we introduce some properties of approximations and these algebraic structures. [ABSTRACT FROM AUTHOR]