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Soft bck/bci-algebras
Abstract
Molodtsov [D. Molodtsov, Soft set theory — First results, Comput. Math. Appl. 37 (1999) 19–31] introduced 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 the theory of BCK/BCI-algebras. The notion of soft BCK/BCI-algebras and soft subalgebras are introduced, and their basic properties are derived.
... Since then, set theoretical aspects of soft sets especially operations of soft sets are studied in [2], [3], [4]. The theory of soft set has also many applications in different kinds of algebraic structures such as groups, semigroups, rings, semirings, near-rings, BCK/BCI algebras and BL-algebras [5][6][7][8][9][10][11][12][13][14]. ...
... 10) Every quasi-ideal of S is semiprime. Proof By Theorem 1, Theorem 2 and Theorem 3, (1) implies (2), (7) implies (10) and (10) implies (1), respectively. Since every soft union bi-ideal S of is a soft union generalized bi-ideal of S, (2) implies (3), (5) implies (6) and (8) implies (9). ...
... Since every soft union bi-ideal S of is a soft union generalized bi-ideal of S, (2) implies (3), (5) implies (6) and (8) implies (9). And since every soft union quasi-ideal of S is a soft union bi-ideal of S, (3) implies (4), (6) implies (7) and (9) implies (10). And by the definition of soft union semiprime, (4) implies (7), (3) implies (6), (2) implies (5). ...
In this paper, certain kinds of regularities of semigroups are studied by correlating soft set theory. Completely, weakly and quasi-regular semigroups are characterized by soft union quasi ideals , soft union (generalized) bi-ideals and soft union semiprime ideals of a semigroup. It is proved that if every soft union quasi-ideal of a semigroup is soft union semiprime, then every quasi-ideal of a semigroup is semiprime and thus, if every quasi-ideal of a semigroup is semiprime, then the semigroup is completely regular. Also, it is obtained that the case when every soft union quasi-ideal (bi-ideal, generalized bi-ideal, respectively) of a semigroup is soft union semiprime is equivalent to the case when every quasi-ideal (bi-ideal, generalized bi-ideal, respectively) of a semigorup is semiprime, where the semigroup is completely semi-group. Similar characterizations are obtained for weakly and quasi-regular semigroups. By these characterizations, we intent to bring a new perspective to the regularities of semigroup theory via soft set theory. Further study can be focused on soft union tri quasi-ideals, soft union bi-quasi ideals, soft union lateral bi-quasi-ideals and soft union lateral tri-quasi ideals of a semigroup. Cite this article as: Sezgin A, Orbay K. Completely weakly, quasi-regular semigroups characterized by soft union Quasi ideals, (generalized) bi-ideals and semiprime ideals. Sigma J Eng Nat Sci 2023;41(4):868−874.
... But mathematical tools may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. However, all of these theories have their own difficulties' (Jun Y.B. 2008). As it was observed in the last work, ' all of these theories have their own difficulties pointed out in (Molodtsov D.A. 1999) § '. ...
... As it was observed in the last work, ' all of these theories have their own difficulties pointed out in (Molodtsov D.A. 1999) § '. The soft set concept was used in (Jun Y.B. 2008) as 'a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches'. And hence, the use of soft sets to the theory of BCK/BCI -algebras ** . ...
... Some basic properties related to the last soft algebraic systems were also derived. † † As an illustration, the above notions of BCK/BCI -algebras were defined as follows (provided there is no ambiguity and for convenience, instead of 'x, y, z X' (Jun Y.B. 2008), equivalently there are used below: 'a,b,c A'). ...
... In what follows, let S(U ) denote the set of all soft sets over U by Cagman et al. [7]. Definition 6. [5] Let (F, A) and (G, B) be two soft sets over U . The intersection of (F, A) and (G, B) is defined to be the soft set (H, C) satisfying the following conditions: ...
... In this case , we write (F, A) ∼ ∩ (G, B) = (H, C). Definition 7. [5] Let (F, A) and (G, B) be two soft sets over a common universe U . Then the union of (F, A) and (G, B) is defined to be a soft set (H, C) satisfying the following conditions: ...
... In this case, we write (F, A) ∼ ∪ (G, B) = (H, C). Definition 8. [5] If (F, A) and (G, B) are two sets over U , then "(F, A) and (G, B)" denoted by (F, A) ...
In this paper, we apply the notion of soft sets to the theory of hyper GR-algebra. Also, we introduce the concept of soft hyper GR-algebras and some properties of soft hyper GR-ideals.
... Ali et al. [6] gave some new notions such as restricted intersection, restricted union, restricted difference, and the extended intersection of soft sets. Jun [24] applied Molodtsov's idea of soft sets to the theory of BCK/BCI-algebras and introduced the notion of soft BCK/BCI-algebras and soft subalgebras, and then investigated their basic properties. Also, the combination of soft sets and rough sets was first explored in [15]. ...
... Also, Soft rings and soft ideals are defined by Acar et al. [1] in (2010) and discussed their basic properties. Since then, some researchers, Jun [24] and Celik et al. [12], have studied other soft algebraic structures and their properties. ...
In this paper, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.
... Ali et al. [6] gave some new notions such as restricted intersection, restricted union, restricted difference, and the extended intersection of soft sets. Jun [24] applied Molodtsov's idea of soft sets to the theory of BCK/BCI-algebras and introduced the notion of soft BCK/BCI-algebras and soft subalgebras, and then investigated their basic properties. Also, the combination of soft sets and rough sets was first explored in [15]. ...
... Also, Soft rings and soft ideals are defined by Acar et al. [1] in (2010) and discussed their basic properties. Since then, some researchers, Jun [24] and Celik et al. [12], have studied other soft algebraic structures and their properties. ...
In this article, we introduce the concept of soft intervals, soft ordering and sequences of soft real numbers, and some of their structural properties are studied. The notion of soft Lebesgue measure on the soft real numbers has been introduced. Also, a correspondence relationship has been established between the soft Lebesgue measure and the classical Lebesgue measure. Furthermore, we have studied some exciting results and relations between the soft Lebesgue measure and the Lebesgue measure of soft real sets.
... Moreover, Aktas and Cagman [11] extended the softness concept to group theory and defined the soft group concept. Furthermore, Feng et al. [12] applied and extended the soft set concept to semirings, Acar [13] initiated the soft rings, and Jun et al. extended the softness concept to BCK/BCI-algebras ( [14][15][16]). Also, Sezgin and Atag€ u n [17] introduced the normalistic soft groups concept, Zhan et al. [18] defined the concept of soft ideal of BL-algebras and Kazanci et al. [19] applied the softness concept to BCH-algebras. ...
... -4 2 -6 π 4 1 -9 1 0 π 5 -7 9 -16 14 Journal of Function Spaces ...
The decision-making technique, launched by Roy and Maji, is considered an effective method to overcome uncertainty and fuzziness in decision-making problems, though, adapting it to reflect the problem parameters' vagueness, as well as multibipolarity, is very difficult. So, in this article, the bipolarity is interpolated into the multivague soft set of order n. This gives a new more generalized, flexible, and applicable extension than the fuzzy soft model, or any previous hybrid model, which is the bipolar-valued multivague soft model of dimension n. Moreover, types of bipolar-valued multivague soft sets of dimension n, as well as some new associated concepts and operations, are investigated with examples. Furthermore, properties of bipolar-valued multivague soft sets of dimension n including absorption, commutative, associative, and distributive properties, as well as De Morgan's laws, are provided in detail. Finally, a bipolar-valued multivague soft set-designed decision-making algorithm, as well as a real-life example, are discussed generalizing the Roy and Maji method.
... Maji et al. [2] instigated a conceptual research of the soft set theory in depth which includes superset and subset of a soft set, operations of union and intersection, null soft set etc. and discussed their properties. Jun and Park [3] and Jun [4] instigated various paths in connection with soft sets applications in the ideal theory of soft BCK/BCI algebras. Many authors explored a few operations on the soft set theory as well. ...
The Molodtsov-conceived concept of soft sets provides a robust mathematical approach for managing uncertainty. This study explores the positive outcomes resulting from the application of soft set theory techniques, paving the way for innovative research methodologies in soft multiplicative ideal theory. The paper delves into the operations that lead to themselves within the realm of undeniable soft fractional ideals. Introducing the notion of soft star-ideals, we illustrate the construction of a complete lattice derived from the collection of all soft star-operations on integral domains. Furthermore, we provide characterizations for pseudo-principal domains, principal ideal domains, and greatest common divisor domains in relation to soft star-operations of undeniable soft fractional ideals.
... In recent years, it has been combined with fundamental mathematical theories such as algebra and topology, resulting in substantial advances. Numerous research have investigated the algebraic features of soft sets, with fundamental contributions from Maji, Aktas, Jun, and Feng [[3], [14], [19], [23]]. Maji et al., [23] proposed new operations on soft sets, while Aktas et al., [3] revised the definitions of soft groups and subgroups, revealing new features. ...
In this paper, the notion of a soft topological transformation group is defined and studied. For a soft topological transformation group, it is proven that a map from a soft topological space onto itself is soft homeomorphism. The collection of all soft homeomorphisms of the given soft topological space onto itself constitutes soft topological group under composition. Subsequently, it is proved that there is a homomorphism between soft topological group and the group structure on the collection of all soft homeomorphisms of given topological space. Subsequently, it is shown that the mapping space Map(Y, Y) is soft Hausdorff and verified that any subspace of the mapping space is soft Hausdorff. Additionally, it is proved that the set of all soft homeomorphisms on Y forms a soft discrete space, soft extremally disconnected space, soft Moscow space and a soft Moscow topological group. Later, it is shown that the map from a soft topological group to a mapping space is soft continuous. Finally, it is proved that a distinct group structure generates distinct collection of all soft homeomorphisms of the specified soft topological space onto itself is a soft isomorphism.
... For this reason, [1], developed soft set theory, a novel method for handling uncertainty. After the work of [1] several researcher's also work on soft set theory [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. ...
The term "soft set", over ⊆ , denoted by (,), is described in this research work as = {(, (): ∈ , () ∈ }, along with a thorough theoretical analysis of the fundamental operations of soft sets, including intersection, extended intersection, union, restricted union, complement and relative complement, Null, and universal soft set. We were able to demonstrate the importance and practical use of soft sets in decision-making through the use of soft "AND-OPERATION" and tabular representation of soft sets. This paper's major goal is to select the top two applicants from the pool of five airline interview by using the notation of the soft AND operation. We identified and demonstrated a few specific properties of how soft set operations work.
... For this reason, [1], developed soft set theory, a novel method for handling uncertainty. After the work of [1] several researcher's also work on soft set theory [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. ...
The term "soft set", over ⊆ , denoted by (,), is described in this research work as = {(, (): ∈ , () ∈ }, along with a thorough theoretical analysis of the fundamental operations of soft sets, including intersection, extended intersection, union, restricted union, complement and relative complement, Null, and universal soft set. We were able to demonstrate the importance and practical use of soft sets in decision-making through the use of soft "AND-OPERATION" and tabular representation of soft sets. This paper's major goal is to select the top two applicants from the pool of five airline interview by using the notation of the soft AND operation. We identified and demonstrated a few specific properties of how soft set operations work.
... Ali et al. [6] demonstrated that De Morgan's laws are valid in the context of SS theory. Jun [7] introduced the idea of soft BCK/BCI-algebras. To determine the relationship that exists between SS and semi rings, Feng et al. [8] characterized soft semi rings and other related concepts. ...
... Imai and Is [7] frst introduced the BCK/BCI-algebra, an algebraic structure of universal algebra, in 1966. Te BCK/BCI-algebraic theory was given a soft set theory treatment by Jun [8]. Later, he used subalgebras and the ideals of BCK/BCI-algebras to apply the idea of hesitant fuzzy soft sets in [9][10][11]. ...
The purpose of this study is to generalize the concept of Q -hesitant fuzzy sets and soft set theory to Q -hesitant fuzzy soft sets. The Q -hesitant fuzzy set is an admirable hybrid property, specially developed by the new generalized hybrid structure of hesitant fuzzy sets. Our goal is to provide a formal structure for the m -polar Q -hesitant fuzzy soft (MQHFS) set. First, by combining m-pole fuzzy sets, soft set models, and Q -hesitant fuzzy sets, we introduce the concept of MQHFS and apply it to deal with multiple theories in B C K / B C I -algebra. We then develop a framework including MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals in B C K / B C I -algebras. Furthermore, we prove some relevant properties and theorems studied in our work. Finally, the application of MQHFS-based multicriteria decision-making in the Ministry of Health system is illustrated through a recent case study to demonstrate the effectiveness of MQHFS through the use of horizontal soft sets in decision-making.
... In [20,22,25], the authors theoretically defined various operations on soft sets and examined the algebraic properties of soft sets. Many authors worked on soft algebraic structures and soft operations [1,11,12,13,14,18,24] after the article [2] publication. In [10], the authors introduced new concepts on the soft sets, which are called soft quotient subgroup and quotient dual soft subgroup. ...
In this paper, firstly four diferent restrictions, alpha-including, alpha-covered, alpha-intersection and alpha-union sets of a soft set over an initial universe U obtained by using a non-empty subset alpha of U are defined and examined in detail. Then, an application of alpha-intersection sets over a soft intersection groups is given. Here, the novel concept of the soft int-subgroup generated by alpha subset U of a group is introduced and some properties are studied. The generators of soft intersections and products of soft int-subgroups are given. Some related properties about generators of soft int-subgroups are investigated and shown by several examples.
... Liu et.al [13] further the investigated isomorphism and fuzzy isomorphism theories of soft rings in [16], respectively. Soft sets were also applied to other algebraic structures such as near-rings [16], Γ-modulus and BCK/BCI-algebras [10]. Bhakat and Das [7] proposed the concept of M-fuzzy subgroups. ...
... Later several author such as Booth [6] and Satyanarayana [19] studied the real theory of Γnear rings. Later Jun.et.al [8,9,10,11] considered the fuzzification of left (respectively right) ideals of Γnear rings. In 1999 Molodtsov [17] proposed an approach for modeling vagueness and uncertainty called soft set theory. ...
The main purpose of this paper is to introduce a basic version of intuitionistic fuzzy soft G-modulo theory, which extends the notion of modules by introducing some algebraic structures in soft set. Finally, we investigate some basic properties of maximal intuitionistic fuzzy soft G-modules.
... Maji et al. [30] presented some definitions on soft sets and based on the analysis of several operations on soft sets Ali et al. [12] introduced several operations of soft sets and Sezgin and Atagün [36] and Ali et al. [13] studied on soft set operations as well. Soft set theory have found its wide-ranging applications in the mean of algebraic structures such as groups [11,37], semirings [18], rings [9], BCK/BCI-algebras [24,25,26], BL-algebras [42], near-rings [35] and soft substructures and union soft substructures [14,38], hemirings [29,43] and so on [18,19,21]. ...
In this paper, semisimple semigroups, duo semigroups, right (left) zero semigroups, right (left) simple semigroups, semilattice of left (right) simple semigroups, semilattice of left (right) groups and semilat-tice of groups are characterized in terms of soft intersection semigroups, soft intersection ideals of semigroups. Moreover, soft normal semigroups are defined and some characterizations of semigroups with soft normality are given.
... In 2007, Aktaş and Ç agman [17] compared soft sets to fuzzy sets and rough sets, gave some basic concepts of soft set theory and defined the concept of soft group. In the following years, many papers were published on soft algebraic structures such as soft intersection semigroups [18][19][20], soft semirings [21], soft rings [22,23], soft near-rings [24], soft intersection Lie algebras [25], soft lattices [26][27][28], soft uni-Abel-Grassmann's groups [29], soft graphs [30,31], soft BCK/BCI-algebras [32][33][34], soft BL-algebras [35]. Moreover, many researchers were studied on the extended models of soft sets such as fuzzy soft sets [36][37][38], intuitionistic fuzzy soft sets [39][40][41][42], T-spherical soft sets [43], neutrosophic soft sets [44,45], three-valued soft sets [46] and N-soft sets [47][48][49][50] and new researches are currently ongoing. ...
In this paper, firstly, we study the decomposition of soft sets in detail. Later, we introduce the concepts of �-upper, �-lower, �-intersection and �-union for soft matrices and present some decomposition theorems. Some of these operations are set-restricted types of existing operations of soft sets/matrices, others are �-oriented operations that provide functionality in some cases. Moreover, some relations of decompositions of soft sets and soft matrices are investigated and the newfound relations are supported with numerical examples. Finally, two new group decision making algorithms based on soft sets/matrices are constructed, and then their efficiency and practicality are demonstrated by dealing with real life problems and comparison analysis.
... Also, Lee and Jun [17] discuss bipolar fuzzy a-ideals in BCK/BCIalgebras. Jun [52][53][54] applied the notion of soft sets to the theory of BCK/BCI-algebras and d-algebras, and introduced the notion of soft BCK/BCIalgebras, soft subalgebras and soft d-algebras, and then described their basic properties. Jun et al. [55] introduced the notion of soft p-ideals and p-ideas of soft BCI-algebras and developed their basic properties. ...
In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of ∈-bipolar fuzzy soft set and q-bipolar fuzzy soft set is introduced based on a bipolar fuzzy set and characterizations for an ∈-bipolar fuzzy soft set and a q-bipolar fuzzy soft set to be bipolar fuzzy soft BCK/BCI-algebras are given. Also, the notion of (∈, ∈ ∨q)-bipolar fuzzy subalgebras and ideals are introduced and characterizes for an ∈-bipolar fuzzy soft set and q-bipolar fuzzy soft set to be a bipolar fuzzy soft BCK/BCI-algebras are established. Some characterization theorems of these (∈, ∈ ∨q)-bipolar fuzzy soft subalgebras and ideals are derived. The relationship among these (∈, ∈ ∨q)-bipolar fuzzy soft subalgebras and ideals are also considered.
... Also, Lee and Jun [17] discuss bipolar fuzzy a-ideals in BCK/BCI-algebras. Jun [51,52,53] applied the notion of soft sets to the theory of BCK/BCI-algebras and d-algebras, and introduced the notion of soft BCK/BCI-algebras, soft subalgebras and soft d-algebras, and then described their basic properties. Jun et al. [54] introduced the notion of soft p-ideals and p-ideas of soft BCI-algebras and developed their basic properties. ...
Toi ntroduce and discuss the properties of bipolar fuzzy soft ideals of BCI/BCK algebras
... Soft set theory has further been developed in many directions, for instance, soft algebraic structures and extensions, hybrid models such as fuzzy soft sets, rough soft sets, soft topology and the semantics of soft sets and decision making (compare [2,[6][7][8][9][10][11][12][13][14][15][16][17][18][19]). Maji et al. [15] presented an elaborated study related to some algebraic operations on soft sets. ...
Soft set theory has evolved to provide a set of valuable tools for dealing with ambiguity and uncertainty in a variety of data structures related to real-world challenges. A soft set is characterized via a multivalued function of a set of parameters with certain conditions. In this study, we relax some conditions on the set of parameters and generalize some basic concepts in soft set theory. Specifically, we introduce generalized finite relaxed soft equality and generalized finite relaxed soft unions and intersections. The new operations offer a great improvement in the theory of soft sets in the sense of proper generalization and applicability.
... Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9][10][11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft K-algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13][14][15][16][17][18]). Jun et al. applied soft set and fuzzy soft set theories to BCK/BCI-algebras in [19,20], respectively, and Akram et al. applied the same theories on K-algebras in [21]. Larimi and Jun introduced the concepts of (∈, ∈∨q)-intuitionistic fuzzy h-ideals of hemiring [22]. ...
In the present paper, using Lukaswize triple-valued logic, we introduce the notion of (α, β)-intuitionistic fuzzy soft ideal of BCK/BCI-algebras, where α and β are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.
... Jun and Park [10] defined the application of SSs in ideal theory. The soft BCK/ BCI algebras are also defined by Jun [11]. Maji et al. [12] defined a utilization of SSs in DM problems. ...
The notion of the T-bipolar soft set defined by Mahmood is a novel approach that can explain bipolarity more effectively than the other notions of bipolar soft sets. But in our daily life there are some situations in which we have to discuss some order between terms like good, satisfactory, excellent, etc. For example, to order the positions of students in a certain class, the teacher ordered the grades like decent, unsatisfactory, really nice, and tremendous. This type of standing is essential to explain the order of students’ positions in a class. In these situations, the T-bipolar soft set fails because it cannot discuss the ordered between terms. So, there is a need for such a notion that can handle these kinds of limitations. Moreover, note that although the concept of lattice ordered soft set is already defined but it discusses only one dimension. The two-dimensional opinions are essential in many decision making problems for ranking some criteria of eligibility and non-eligibility options. Therefore, to cover these mentioned drawbacks, in this article, we have elaborated a new idea of lattice (anti-lattice) ordered T-bipolar soft sets that can consider the order among parameters and it is a more valuable structure for certain situations in decision-making problems. Therefore, when we define the ranking among the set of parameters, the concept of decision making becomes very helpful under the conception of lattice (anti-lattice) ordered T-bipolar soft sets. Moreover, basic operations are defined based on introduced notions like restricted union and restricted intersection. Furthermore, the extended union and the extended intersection of introduced notions have been established. To use the introduced notion to decision making problems, we have established the application in this regard. Furthermore, we have proposed a TOPSIS method that can strengthen the study of the introduced notion and verify the results obtained from introduced notions as well. The comparative analysis of introduced work show the superiority and effectiveness of established work than that of existing notions.
... Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9][10][11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft K-algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13][14][15][16][17][18]). Jun et al. applied soft set and fuzzy soft set theories to BCK/BCI-algebras in [19,20], respectively, and Akram et al. applied the same theories on K-algebras in [21]. Larimi and Jun introduced the concepts of (∈, ∈∨q)-intuitionistic fuzzy h-ideals of hemiring [22]. ...
In the present paper, using Lukaswize triple-valued logic, we introduce the notion of (α, β)-intuitionistic fuzzy soft ideal of BCK/BCI-algebras, where α and β are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.
... Aktas and Cagman described soft groups [23]. Jun connected SSs with ideal theory ( [24,25]). Maji et al. [26] worked on IFSSs. Razak and Mohamad [27] worked on decision-making regarding fuzzy soft sets (FSSs) in connection with SSs [28]. ...
The major advantage of this proposed work is to investigate roughness of intuitionistic fuzzy subsemigroups (RIFSs) by using soft relations. In this way, two sets of intuitionistic fuzzy (IF) soft subsemigroups, named lower approximation and upper approximation regarding aftersets and foresets, have been introduced. In RIFSs, incomplete and insufficient information is handled in decision-making problems like symptom diagnosis in medical science. In addition, this new technique is more effective as compared to the previous literature because we use intuitionistic fuzzy set (IFS) instead of fuzzy set (FS). Since the FS describes the membership degree only but often in real-world problems, we need the description of nonmembership degree. That is why an IFS is a more useful set due to its nonmembership degree and hesitation degree. The above technique is applied for left (right) ideals, interior ideals, and bi-ideals in the same manner as described for subsemigroups.
... Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9][10][11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft K-algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13][14][15][16][17][18]). Jun et al. applied soft set and fuzzy soft set theories to BCK/BCI-algebras in [19,20], respectively, and Akram et al. applied the same theories on K-algebras in [21]. Larimi and Jun introduced the concepts of (∈, ∈∨q)-intuitionistic fuzzy h-ideals of hemiring [22]. ...
In the present paper, using Lukaswize triple-valued logic, we introduce the notion of (α, β)-intuitionistic fuzzy soft ideal of BCK/BCI-algebras, where α and β are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.
... Jun [10] introduced the concept of "soft BCK/BCI-algebras" by applying soft sets to the theory of BCK/BCI-algebras. ...
... Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9][10][11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft K-algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13][14][15][16][17][18]). Jun et al. applied soft set and fuzzy soft set theories to BCK/BCI-algebras in [19,20], respectively, and Akram et al. applied the same theories on K-algebras in [21]. Larimi and Jun introduced the concepts of (∈, ∈∨q)-intuitionistic fuzzy h-ideals of hemiring [22]. ...
This article presents the notion of soft 2-normed linear space (NDLS) and extends it to soft n-NDLS, providing a versatile framework beyond traditional soft NDLS and n-normed spaces (NDS). Established results in soft NDLS are adapted to soft n-NDLS, bolstered by practical examples.
In this study, we describe soft parts, named soft 0-symmetric and soft constant parts, of soft intersection near-rings and soft union near-rings, and obtain their fundamental features. We explore the relations between the parts of near-rings and soft parts of soft intersection near-rings and soft union near-rings, and we give some applications of these parts to soft sets. Additionally, soft intersection (union) product of soft intersection (union) near-ring are introduced and applied on soft parts of soft intersection near-rings and soft union near-rings, respectively.
In this paper, a different approach is used to define a cartesian product on soft sets. This method processes both alternatives and parameters. The notion of the cartesian product is then used to define the idea of a soft relation. The concepts of reflexion, symmetry and transition are defined on the soft relation. Some properties are investigated. Also, the soft function notion is introduced. Various instances are provided as the key characteristics of the structures that are being presented are analyzed. Finally, an application is presented by building a decision making algorithm on the soft relation.
Some kinds of pseudo valuations such as positive implicative pseudo valuation, (weak) implicative pseudo valuation, and commutative pseudo valuation of various types are introduced. Several examples, properties and characterizations of them are given as well. The relationships between them and the substructures of hyper BCK -algebras are investigated, too. Finally, by giving various examples and theorems, the relationships among the proposed pseudo valuations are investigated and characterized, especially in hyper BCK -algebras with three elements.
Balamurugan et al. introduced the notion of anti-intuitionistic fuzzy soft ideals in BCK/BCI-algebras [5] and Muhiuddin et al. introduced Anti-Intuitionistic Fuzzy Soft a-Ideals applied to BCI-Algebras [13]. In this article, the notion of anti-intuitionistic fuzzy soft modules in BCK/BCI-algebras has been introduced and some of their properties have been discussed.
Ideal concepts are discussed in many mathematical applications. Various author has been studied and analytical in different ways. In this article, the idea of bipolar fermatean uncertainty sub algebra’s in terms of R-ideals is planned. Also the correlation among bipolar fermatean uncertainty soft ideal and bipolar fermatean uncertainty soft R-ideals is expressed some interesting ideas also analyzed.
A new structure, so called soju structure, is introduced by combining intuitionistic fuzzy set and soft set, which is applied to BCK/BCI-algebras. The notions of soju subalgebra and soju ideal in BCK/BCI-algebras are introduced, and related properties are investigated. A strong soju subalgebra in BCI-algebras is introduced, and a condition for a soju subalgebra to be strong is provided. Relations between soju subalgebra and soju ideal are discussed, and conditions for a soju structure to be a soju ideal in a BCK-algebra are considered. Characterizations of soju subalgebra and soju ideal are considered, and the homomorphic image and preimage of soju subalgebra is discussed.
Near soft sets are a very successful mathematical model that has been used in order to express the decision-making process for uncertainty in a more ideal way, especially in recent years. The purpose of this paper is to contribute to the theoretical studies on near soft topological spaces. In addition, it presents basic concepts and constructs that will form the basis for a near theoretical set-up of near soft topological spaces. These concepts and structures include sub near soft set, near soft subspaces of a near soft topological space and near soft -spaces for . The important aspects of the paper are discussed, especially by examining the definitions and properties given.
In this paper, we discuss the properties of interval valued bi-cubic homology fuzzy soft sub modules and its arbitrary intersections. Also the level subset of homology soft modules indexed with its interval valued fuzzy soft set has been analysed. Finally,we proved that the inverse image of an interval valued bi-cubic homology fuzzy soft modules is also an interval valued bi-cubic homology fuzzy soft modules.
In this expose, on endeavors equipped on the way to achieve comprehension of the arithmetical character of -fuzzy soft sub hemi rings of a hemi ring.
In this paper, we have studied the algebraic operations of anti S-fuzzy soft sets to establish their basic properties. We have discussed different algebraic structures of anti S-fuzzy soft sets under the restricted and extended operations of union and intersection in a comprehensive manner. Logical equivalences have also been made in order to give a complete overview of these structures.
The purpose of this research is to interpolate bipolarity into the defnition of the vague soft set. Tis gives a new more applicable, fexible, and generalized extension of the soft set, the fuzzy soft set, or even the vague soft set, which is the bipolar vague soft set. In addition, types of bipolar vague soft sets, as well as some new related concepts and operations are established with examples. Moreover, properties of bipolar vague soft sets including absorption, commutative, associative, distributive, and De Morgan's laws are discussed in detail. Furthermore, a bipolar vague soft set-designed decision-making algorithm is provided generalizing Roy and Maji method. Tis allows making more efective decisions to choose the optimal alternative. Finally, an applied problem is introduced with a comparative analysis to illustrate how the proposed algorithm works more successfully than the previous models for problems that contain uncertain ambiguous data.
In recent years, soft sets have become popular in various fields. For this reason, many studies have been carried out in the field of algebra. In this study, soft intersection k-ideals are defined with the help of a semiring, and some algebraic structures are examined. Moreover, the quotient rings are defined by k-semiring. Isomorphism theorems are examined by quotient rings. Finally, some algebraic properties are investigated by defining soft intersection maximal k-ideals.
The concept of a positive implicative makgeolli ideal in BCK-algebras is introduced, and its properties are investigated. The relationship between a makgeolli ideal and a positive implicative makgeolli ideal is established. The conditions under which a makgeolli ideal can be a positive implicative makgeolli ideal are explored. Characterizations of a positive implicative makgeolli ideal are discussed, and the extension property for a positive implicative makgeolli ideal is established.
We introduce spherical fuzzy neutrosophic cubic graph and single-valued neutrosophic spherical cubic graphs in bipolar setting and discuss some of their properties such as Cartesian product, composition, m-join, n-join, m-union, n-union. We also present a numerical example of the defined model which depicts the advantage of the same.Finally, we define a score function and minimum spanning tree algorithm of an undirected bipolar single-valued neutrosophic spherical cubic graph with a numerical example.
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In this paper, we show that the inappropriateness of Example 3.10 and the error of Theorem 4.7 in [M.Balamurugan, G.Balasubramanian, C.Ragavan. Intuitionistic Fuzzy Soft Ideals in BCK/BCI-algebras. Materials today: proceedings 16(2019)496-503.], then give a more perfect example and a counterexample respectively.
Molodtsov’s soft set is a novel mathematical approach to address models containing uncertainty. The algebraic structures of this set approach are prominent issues. The main objective of this paper is to contribute algebraic structures in the soft set theory. Relatedly, the notions of soft intersection near-field and soft union near-field, and their fundamental properties are introduced. Also, some findings and results related to the emerging soft algebraic structures are included.
Suppose that all of C∞ functions f1,…, fk have the zero property. We give a necessary and sufficient condition for their product to have the same property This is a generalization of Bochnak’s result ([1]).
The fuzzification of (normal) B-subalgebras is considered, and some related properties are investigated. A characterization of a fuzzy B-algebra is given.
The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future.
We consider the fuzzification of the notion of implicative hyper
BCK-ideals, and then investigate several properties.
Using the concept of level subsets, we give a
characterization of a fuzzy implicative hyper BCK-ideal. We state
a relation between a fuzzy hyper BCK-ideal and a fuzzy implicative
hyper BCK-ideal. We establish a condition for a fuzzy hyper
BCK-ideal to be a fuzzy implicative hyper BCK-ideal. Finally, we
introduce the notion of hyper homomorphisms of hyper
BCK-algebras, and discuss related properties.
The fuzzy setting of a left h-ideal in a hemiring is constructed, and basic properties
are investigated. Using a collection of left h-ideals of a hemiring S, fuzzy left h-ideals of
S are established. The notion of a finite valued fuzzy left h-ideal is introduced, and its
characterization is given. Fuzzy relations on a hemiring S are discussed.
The fuzzy setting of strong implicative hyper BCK-ideals in hyper BCK-algebras is considered, and some of their properties are investigated. Relations among fuzzy strong hyper BCK-ideals, fuzzy implicative hyper BCK-ideals, and fuzzy strong implicative hyper BCK-ideals are given. A characterization of a fuzzy strong implicative hyper BCK-ideal is provided. The hyper homomorphic preimage of a fuzzy strong implicative hyper BCK-ideal is discussed.
Prove that the notion of positive implicative BCI-algebras coincides with that of weakly positive implicative BCI-algebras, thus the whole results in the latter are still true in the former, in particular, one of these results answers definitely the first half of J. Meng and X.L. Xin’s open problem: Does the class of positive implicative BCI-algebras form a variety? The second half of the same problem is: What properties will the ideals of such an algebra have? Here, some further properties are obtained.
In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.
In this paper, we focus our discussion on the parameterization reduction of soft sets and its applications. First we point out that the results of soft set reductions offered in [1] are incorrect. We also observe that the algorithms used to first compute the reduct-soft-set and then to compute the choice value to select the optimal objects for the decision problems in [1] are not reasonable and we illustrate this with an example. Finally, we propose a reasonable definition of parameterization reduction of soft sets and compare it with the concept of attributes reduction in rough sets theory. By using this new definition of parameterization reduction, we improve the application of a soft set in a decision making problem found in [1].
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
In this paper, we apply the theory of soft sets to solve a decision making problem using rough mathematics.
In this paper we discuss quotient implicative algebras induced by fuzzy implicative filters, and establish the fuzzy fundamental homomorphism theorem for some implicative algebras.
The fuzzification of subinclines (ideals) in inclines are considered, and some related properties are investigated. We also state the product of fuzzy subinclines (ideals), and the projections of fuzzy subinclines (ideals). We discuss fuzzy relations, fuzzy characteristic subinclines (ideals).
We state the relation between fuzzy prime ideals and invertible fuzzy ideals in commutative BCK-algebras.
We fuzzify the concept of positive implicative filters and associative filters in lattice implication algebras. We prove that (i) every fuzzy positive implicative filter is a fuzzy implicative filter, and (ii) every fuzzy associative filter is a fuzzy filter. We provide equivalent conditions for both fuzzy positive implicative filter and fuzzy associative filter.
The purpose of this paper is description of properties of fuzzy BCC-ideals in BCC-algebras and its images. We describe also an extension of a fuzzy BCC-ideal μ of a given BCC-subalgebra S of X to a fuzzy BCC-ideal of X such that μ and have the same image.
The concept of a fuzzy annihilator in a commutative BCK-algebra will be introduced and then we investigate some basic properties. Using this notion, we define an involutory (resp. invertible) fuzzy ideal. We prove that (i) every bounded implicative BCK-algebra is an involutory fuzzy BCK-algebra, and (ii) every categorical commutative BCK-algebra is an invertible fuzzy BCK-algebra. Let X be an involutory and invertible fuzzy BCK-algebra and let FI(X) denote the set of all fuzzy ideals of X. We show that (iii) (FI(X),∪,∩) is a distributive lattice, and (iv) (FI(X),ι,∪,∩,⊛) is a quasi-Boolean algebra.
Characterizations of fuzzy filters in MTL-algebras are given. The fuzzy filter generated by a fuzzy set is considered. The notion of Boolean fuzzy filters and MV-fuzzy filters are introduced and related properties are investigated. A condition for a fuzzy filter to be Boolean is provided. A characterization of a Boolean fuzzy filter is given. A congruence relation on a MTL-algebra induced by a fuzzy filter is established, and we show that the set of all congruence relations induced by a fuzzy filter is a completely distributive lattice.
The fuzzy setting of a left h-ideal in a hemiring is constructed, and basic properties are investigated. Using a collection of left h-ideals of a hemiring S, fuzzy left h-ideals of S are established. The notion of a finite valued fuzzy left h-ideal is introduced, and its characterization is given. Fuzzy relations on a hemiring S are discussed.
On fuzzy B-algebras, Czechoslovak Math
- Y B Jun
- E H Roh
- H S Kim
Y.B. Jun, E.H. Roh, H.S. Kim, On fuzzy B-algebras, Czechoslovak Math. J. 52 (127) (2002) 375–384.
The parametrization reduction of soft sets and its applications
- Chen