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Applications of soft sets in ideal theory of BCK/BCI-algebras
Abstract
The notion of soft ideals and idealistic soft BCK/BCI-algebras is introduced, and several examples are given. Relations between soft BCK/BCI-algebras and idealistic soft BCK/BCI-algebras are provided. The intersection, union, “AND” operation, and “OR” operation of soft ideals and idealistic soft BCK/BCI-algebras are established.
... Conversely, let M (X,E) := (M E , G E , ξ) be a makgeolli structure on (X, E) that satisfies (17). Let a, b, c ∈ E and x, y, z ∈ X be such that a ↬ b ≤ c and x * y ≤ z. ...
... Let a, b, c ∈ E and x, y, z ∈ X be such that a ↬ b ≤ c and x * y ≤ z. Then ((a ↬ (2) and (17). Hence M (X,E) := (M E , G E , ξ) is a makgeolli ideal of (X, E) by Lemma 2. Since (((a ↬ b) ↬ b) ↬ ((a ↬ b) ↬ b)) ↬ 0 = 0 and (((x * y) * y) * ((x * y) * y)) * 0 = 0 for all a, b ∈ E and x, y ∈ X, It follows from (9) and (17) that ...
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.
... (1) In [2], Theorem 4.6, Corollary 4.7, Theorem 5.8, and Corollary 5.9 are incorrect as they dealt with an incorrect operation (intersection) of soft sets. ...
This note is written to show that some results of [H. Aktas and N. Cagman, Information Sciences, 177 (2007), 2726-2735] are incorrect. Consequently , as this paper is the first paper on soft algebraic structures, some of the results in related papers are incorrect. In 1999, Molodtsov [6] introduced the concept of soft set and provided some applications of the new theory. The new theory grabbed the attention of many mathematicians and computer scientists especially after the publishing of Maji et al. paper [5] in 2003 that discussed some operations on soft sets such as intersection , union, complement, etc. Unfortunately, some of the defined operations in [5] are incorrect and this was pointed by Molodtsov [7] in 2018. Where he provided a detailed description about the correct operations and relations of soft sets. In 2007, Aktas and Cagman [1] introduced soft groups and soft subgroups. All the results in their paper related to intersection and union of soft groups (soft subgroups) are incorrect as they depend on the following definitions and it was shown by Molodtsov [7] that they are incorrect.
... 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.
... 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]. ...
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.
... The application of soft set theory to decision theory problems was proposed in [11]. Subsequent considerations expanded the exploration of soft sets into algebra and topology, as evidenced by works such as [12][13][14][15]. This expansion facilitated the development of multicriteria decision-making methods, notably employed in medical applications for diagnostic purposes. ...
In recent years, mathematical theories and tools specifically aimed at solving complex real-world problems and the consequent representation of models characterized by a certain degree of ambiguity, uncertainty and vagueness have assumed an increasingly important role in the landscape of pure and applied science disciplines. Among them, particularly Soft Set theory and Neutrosophic Set theory have revolutionized our ability to understand and model physical phenomena more accurately and they have consequently seen a growth in interest and research efforts as well as scientific production in their respective fields.
... Muhiuddin and Al-Roky expanded on this approach by merging cubic and soft-set concepts [15]. Their research entailed a thorough analysis of the characteristics and applications of cubic soft sets, particularly with B-algebras [16]. Exploration of Subalgebras Authors such as Senapati and Iqbal investigated cubic soft subalgebras and cubic soft ideals, studying their characterizations and linkages within the context of B-algebra [17]. ...
In this paper, we introduce the concepts of cubic soft (CS) algebra, CS o-subalgebra, and CS ideals within the framework of B-algebra. We provide comprehensive characterizations of these new structures, elucidating their unique properties and interrelationships. Specifically, we present detailed conditions under which a CS subalgebra can be classified as a closed CS ideal. Our analysis explores the intricate relationships among closed cubic soft ideals, cubic soft subalgebras, and cubic soft o-subalgebras. By doing so, we aim to provide a deeper understanding of how these structures interact and coexist within the broader context of B-algebra The findings offer significant insights into the application and theoretical underpinnings of cubic soft sets in algebraic systems, contributing to the ongoing evolution of fuzzy set theory and its applications in various mathematical domains. Our work not only broadens the scope of B-algebra but also enhances its utility in solving complex problems where traditional algebraic approaches may fall short. Through this exploration, we seek to advance the field and open new avenues for research and practical applications in mathematical sciences.
... Aktas and Cagman [15] started the use of in algebra. In BCK/BCI algebra, Jun and Park [16] discussed soft ideal theory. Moreover, Ali et al. [17] introduced algebraic notions of based on new operations. ...
The release of harmful materials into the environment is called pollution and the harmful materials are called pollutants. There are four basic categories of pollution: land, water, noise, and air pollution. All forms of pollution often have severe consequences on human health as well as the environment and wildlife. There are certain decision-making scenarios like the phenomenon of voting where we have to utilize the third grade called abstinence grade along with membership grade and non-membership grade. Many remarkable fuzzy structures like the intuitionistic fuzzy set, Pythagorean fuzzy set and q-rung orthopair fuzzy set can never discuss abstinence grades that show their flaws. Moreover, we can observe that the parametrization tool is a remarkable instrument used in soft set theory and all above-mentioned structures fail to cover the parametrization as well. Moreover, Einstein operations comprise Einstein product and Einstein sum, which serve as excellent substitutes for algebraic product and algebraic sum. So keeping in view the characteristics of the parametrization tool, the more advanced structure of the picture fuzzy soft set and Einstein operational rules, in this article, we have established Einstein operational laws for picture fuzzy soft numbers. Moreover, we have elaborated the basic notion of Einstein-weighted average operators and Einstein-weighted geometric aggregation operators. Furthermore, we have discussed the basic properties of these introduced notions. Moreover, we have discussed the algorithm for the application of these aggregation operators in the identification of types of pollution that mostly affect the environment. We have provided a comparison of these introduced works for the superiority of these introduced conceptions.
... Fuzzy soft sets are an extension introduced by Maji et al. [12] and provide a more flexible approach to handling uncertainty. Since then there has been a rapid growth of interest in soft sets and their various applications to algebraic systems [13][14][15][16][17][18][19][20], data analysis [21], and decision making under uncertainty [22][23][24][25][26]. ...
The concept of the hybrid structure, as an extension of both soft sets and fuzzy sets, has gained significant attention in various mathematical and decision-making domains. In this paper, we delve into the realm of hemirings and investigate the properties of hybrid h-bi-ideals, including prime, strongly prime, semiprime, irreducible, and strongly irreducible ones. By employing these hybrid h-bi-ideals, we provide insightful characterizations of h-hemiregular and h-intra-hemiregular hemirings, offering a deeper understanding of their algebraic structures. Beyond theoretical implications, we demonstrate the practical value of hybrid structures and decision-making theory in handling real-world problems under imprecise environments. Using the proposed decision-making algorithm based on hybrid structures, we have successfully addressed a significant real-world problem, showcasing the efficacy of this approach in providing robust solutions.
... In the following years, many authors have studied the soft algebraic structures such as soft semirings (Feng et al. 2008), soft rings (Acar et al. 2010;Kamacı 2019, 2021), soft ideals (Kamacı 2020;Sezer 2012;Sezer etal. 2015), soft BCK/BCI/BCH algebras (Jun and Park 2008;Jun et al. 2009;Kazancı et al. 2010), soft Lie algebras (Akram 2013;Akram and Feng 2013), soft near-rings ) and soft lattices (Karaaslan et al. 2012;Susanta etal. 2017;Zhan and Xu 2011). ...
In this paper, two new uncertainty modeling concepts, namely strait soft set and strait rough set, between the structures of rough sets and soft sets, which will bring new perspectives to both theoretical and practical aspects, are presented. A reduction method of alternatives is given using strait soft sets. Another convenience arising from the structure of strait soft sets is that they allow the parameters to be combined. Thus, the fusion of parameters is defined and its use in soft set operations is demonstrated. Strait rough sets naturally contain the characteristics of the rough sets and also allow the parameters to be characterized. The strait soft set and strait rough set are supported by many examples and comparisons. In addition, a new decision-making approach based on the strait soft set and strait rough set is proposed and then followed by real-life applications to illustrate the computational processes.
... 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.
... By introducing soft intersection, union products and soft characteristic functions, made a new approach to the classical ring theory via the soft set theory, with the concepts of soft union rings, ideals and bi-ideal. Jun et.al [11] applied intersectional soft sets to BCK/BCI-algebras [10] and obtained many results. ...
... Also, Asif and coauthors [2] determined fully prime double-framed soft ordered semigrouops. For further reading on ordered semigroups, soft sets and double-framed soft sets we refer the reader to references ( [8,10,12,13,14,15,16,17,18,19,22,23,26,27,28,30]). ...
... Feng et al. [11] suggested approach of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings, soft semiring homomorphisms and their characteristics. Approach of soft BCK/BCI-algebras and soft subalgebras are presented by Jun [13,14,15]. Maji et al. [20] suggested this approach. ...
Molodtsov was a father of soft set approach. We can’t easily settle the membership degree in some practical application. So it must be much better to describe interval-valued data instead of explaining membership degree. In this paper, we introduce the latest approach of the interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set models. This approach successfully follows distributive, associative and DeMorgan’s laws as well. In the end, a decision problem is solved by this approach.
... in 1999, came forward with a novel mathematical tool (named, soft sets) to analyze the uncertain situations, where complete information are not available and accentuated the need of usage of soft sets in various fields with interesting examples. In the second phase, after Molodtsov, the names of P. K. Maji Atagun are of vital importance in the development of soft logic to make it able enough to be used in applied fields, see [3,5,6,8,9,12,13,14,15,16,22,23,25,26,29]. The idea of semirings have been focal point of study by mathematicians and has set its worth in information sciences. ...
In this paper we introduce the notions of soft intersectional ternary subsemirings and soft intersectional ideals in ternary semirings. We also discuss some basic results associated with these notions. In the last part of the paper we characterize regular and weakly regular ternary semirings by their soft intersectional ideals.
... 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.
... Soft set theory, when combined with fuzzy set theory (Zadeh 1965) Yang et al. (2013) and Peng et al. (2015), and also used in forecasting problems as in Xiao et al. (2009). There are also some applications of soft set theory in algebraic structures as in Acar et al. (2010), Aktaş and Çagman (2007) and Jun and Park (2008). When soft set theory is combined with rough set theory (Pawlak 1982), we get new approximation spaces with interesting properties (Shabir et al. 2013). ...
In this paper, we study the notion of equivalent sets in soft topological spaces. We investigate the properties of soft equivalent sets in soft topological spaces under different circumstances. Specifically, we look at how soft equivalent sets behave with respect to different soft topological separation axioms.
... 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 the field of algebra theory, Aktas and Cagman [1] explained the notion of soft groups as a parametrized family of subgroups of a group. Following this definition, many researchers ( [2], [4]- [7], [10], [16]) studied the algebraic properties of soft sets. Cagman et al. [3] introduced a different group structure on a soft set based on the inclusion relation and intersection of sets. ...
... 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.
... 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. ...
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.
... 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.
... Also, many researchers have applied the theory of soft sets to data science, and new mathematical structures have been İ . Altıntaş et al. continued to be established on soft sets (Chen et al. 2005;Pei and Miao 2005;Aktaş and Çagman 2007;Zou and Xiao 2008;Jun and Park 2008;Feng et al. 2010;Çetkin and Aygün 2016;Tahat et al. 2018;Kandemir 2018;Terepeta 2019;Alcantud 2020). ...
This paper is an introduction to soft partial metric spaces. The aim is to create a soft topological model for a programming language described as a soft logic system, like in classical partial metric studies. Since the soft metric spaces have Hausdorff properties, they are not useful in examining non-Hausdorff soft topologies. This paper proposes a generalized soft metric for non-Hausdorff soft topologies and a new approach that guides how to expand soft metric implements like the Banach theorem to such topologies.
... Fuzzification was applied to BCK/BCI-algebras. For example, Jun et al. [11,12] investigated soft ideals of BCK/BCI-algebras, Al-Masarwah and Ahmad [13,14] discussed multipolar fuzzy ideals of BCK/BCI-algebras. Some applications of BCK-algebras can be found, e.g., in [15,16]. ...
In this paper, we apply the concept of linear Diophantine fuzzy sets in BCK/BCI-algebras. In this respect, the notions of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are introduced and some vital properties are discussed. Additionally, characterizations of linear Diophantine fuzzy subalgebras and linear Diophantine fuzzy (commutative) ideals are considered. Moreover, the associated results for linear Diophantine fuzzy subalgebras, linear Diophantine fuzzy ideals and linear Diophantine fuzzy commutative ideals are obtained.
... The idea of soft semirings is introduced by Feng et al. [9]. Jun and Park [10] defined the application of SSs in ideal theory. The soft BCK/ BCI algebras are also defined by Jun [11]. ...
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.
... Also, Asif and coauthors [3] determined fully prime double-framed soft ordered semigrouops. For further reading on ordered semigroups, soft sets and double-framed soft sets we refer the reader to references ( [8,10,12,13,14,15,16,17,18,19,22,23,26,27,28,30]). ...
The notions of a prime (strongly prime, semiprime, ir-
reducible, and strongly irreducible) double-framed soft
bi-ideals (brie
y, prime, (strongly prime, semiprime, ir-
reducible and strongly irreducible) DFS bi-ideals) in or-
dered semigroups are introduced and related properties
are investigated. Several examples of these notions are
provided. The relationship between prime and strongly
prime, irreducible and strongly irreducible DFS bi-ideals
are considered and characterizations of these concepts are
established. The Characterizations of regular and intra-
regular ordered semigroups in terms of these notions are
studied
... Since then, a lot of research has emerged on the theory of BCK/BCI-algebras, with a particular focus on the ideal theory of BCK/BCI-algebras. Different types of ideals were examined in various methods in BCK/BCI-algebras (see, for example, [15,16,27,28,31]). ...
In this paper, by combining the notions of m-polar fuzzy structures and interval-valued m-polar fuzzy structures, the notion of m-polar cubic structures is introduced and applied on the ideal theory of BCK/BCI-algebras. In this respect, the notions of m-polar cubic subalgebras and m-polar cubic (commutative) ideals are introduced and some essential properties are discussed. Characterizations of m-polar cubic subalgebras and m-polar cubic (commutative) ideals are considered. Moreover, the relations among m-polar cubic subalgebras, m-polar cubic ideals and m-polar cubic commutative ideals are obtained.
... Since then, a large number of studies have been published concerning the theory of BCK/BCI-algebras. Many authors, especially Liu et al. [4], Khalid and Ahmad [5], Jun et al. [6][7][8], Muhiuddin et al. [9,10], and Al-Masarwah and Ahmad [11], studied different aspects of BCK/BCI-algebras based on ideal theory. e idea of quasi-coincidence of a fuzzy point with a fuzzy set, as stated in [12], was fundamental in the development of various types of fuzzy subgroups, known as (α, β)-fuzzy subgroups, as defined by Bhakat and Das in [13]. ...
In this paper, we introduce the notion of ∈ , ∈ ∨ κ ∗ , q κ -fuzzy q -ideals of BCI-algebras to propose a more general form of fuzzy q -ideals of BCI-algebras. We prove that ∈ , ∈ ∨ q -fuzzy q -ideals and ∈ ∨ κ ∗ , q κ , ∈ ∨ κ ∗ , q κ -fuzzy q -ideals are ∈ , ∈ ∨ κ ∗ , q κ -fuzzy q -ideals, but the converse assertion is not valid and examples are given to support this. It is proved that every ∈ , ∈ ∨ κ ∗ , q κ -fuzzy q -ideal is an ∈ , ∈ ∨ κ ∗ , q κ -fuzzy ideal, but the converse need not be true in general and an example is provided. In addition, correspondence between ∈ , ∈ ∨ κ ∗ , q κ -fuzzy q -ideals and q -ideals of BCI-algebras is considered.
... In [7], Maji et al. present an application of SSs in a decisionmaking problem. Jun et al. [8] gave the relations between soft BCK/BCI-algebras and idealistic soft BCK/BCI algebras. Chen et al. [9] focus on the parameterization reduction of SSs and their applications. ...
Interval-valued intuitionistic neutrosophic soft sets, Quadripartitioned neutrosophic soft sets, Interval-valued intuitionistic quadripartitioned neutrosophic soft sets Abstract − Molodtsov introduced a soft set (SS) to model uncertainty parametrically and Chaterjee et al. proposed the notion of quadripartitioned neutrosophic set (QNS) by dividing indeterminacy into two independent components, namely contradiction () and unknown (). Afterward, by combining the SS and QNS, a new concept known as quadripartitioned neutrosophic soft set (QNSS) is introduced. In relation to the concept of QNSS, another concept called interval-valued intuitionistic quadripartitioned neutrosophic soft set (in short IVIQNSS) is established to handle more complex indeterminate information parametrically with the restricted conditions. This paper aims to further generalize the existing soft models by introducing an IVIQNSS to explore another kind of imprecise knowledge. The IVIQNSS model can be viewed as a more flexible and powerful framework to encounter indeterminacy parametrically with , , , and as dependent interval quadripartitioned neutrosophic components where , , , ⊆ [0,1] such that sup + sup ≤ 1 , and sup + sup ≤ 1. So, by using the IVIQNSS framework we are capable to address the indeterminate, inconsistent, and incomplete information more accurately. Different operations such as complement, AND, OR, union, intersection, etc. are defined on IVIQNSSs. Furthermore, an algorithm is constructed to solve decision-making (DM) problems based on IVIQNSS. Finally, an illustrative example is executed to validate the proposed study. Subject Classification (2020): 03E72, 03F55.
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.
In recent years, soft sets have been widely used in many important decision-making real-life problems. In this paper, observing the usage of soft sets in such kind of vital problems, we have introduced the bijective-unitary bijective soft rings. Firstly, we have defined and exemplified a bijective soft ring and a unitary bijective soft ring. Moreover, we have presented some applications of bijective soft rings. We have shown the usage of bijective soft rings in coding theory. In this context, we have observed that by obtaining a bijective soft ring over a finite ring, we have a coding matrix to encode a given set of messages. Besides these applicable results, we have also obtained some relations between bijective soft and classical rings.
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.
In this paper, we introduce the concept of S-topological BE-algebras (STBE-algebras) as a generalization of the concept of topological BE-algebra by using the concepts of semi-open sets and prove some of its properties. We also introduce the topological concepts, open sets, closed sets, interior and closure in STBE-algebras and arrive at the topological properties of STBE-algebras.
This is an Open Access Journal / article distributed under the terms of the Creative Commons Attribution License (CC BY-NC-ND 3.0) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All rights reserved. In this paper, we introduce the concept of left and right maps in a self distributive STBE-Algebras. From the collection of subsets of the right maps of a self distributive STBE-Algebra, we obtain a special topology, called RS− Topology on the class of all right maps. Then we construct the set of all right maps as a STBE− Algebra under a binary operation ⊛ and the topology ′ .
Soft sets were introduced as a means to study objects that are not defined in an absolute way and have found applications in numerous areas of mathematics, decision theory, and in statistical applications. Soft topological spaces were first considered in Shabir and Naz ((2011). Computers & Mathematics with Applications 61 (7) 1786–1799) and soft separation axioms for soft topological spaces were studied in El-Shafei et al. ((2018). Filomat 32 (13) 4755–4771), El-Shafei and Al-Shami ((2020). Computational and Applied Mathematics 39 (3) 1–17), Al-shami ((2021). Mathematical Problems in Engineering 2021 ). In this paper, we introduce the effective versions of soft separation axioms. Specifically, we focus our attention on computable u-soft and computable p-soft separation axioms and investigate various relations between them. We also compare the effective and classical versions of these soft separation axioms.
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.
This work aims to introduce and discuss two new classes of separation properties namely, soft generalized R0 and R1
in a soft generalized topological space defined on an initial universe set, by using the notions of soft g-open sets and soft g-
closure operator. We investigate some of their properties and characterizations. We further, investigate the relationships between
different generalized structures of soft topology, providing some illustrative examples and results. Additionally, we present
connections between these separation properties and those in some generated topologies. Furthermore, we show that being
SGRi, i = 0, 1 are soft generalized topological properties.
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.
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.
The present investigation is concerned with the estimation of the upper bound to the H4 (p) Hankel determinant for a subclass of p-valent functions in the open unit disc E={z:|z|<1}. This work will motivate the researchers to work in the direction of investigation of fourth Hankel determinant for several other subclasses of univalent and multivalent functions.
Soft separation axioms and their properties are popular topic in the research of soft topological spaces. Two types of separation axioms Ti-I and Ti-II (i = 0, 1, ⋯ , 4) which take single point soft sets and soft points as separated objects have been given in [18] and [30] respectively. In this paper we show that a soft T0-II(T1-II, T2-II, and T4-II respectively) space is a soft T0-I(T1-I, T2-I, and T4-I respectively) space, if the initial universe set X and the parameter set E are sets of two elements. Some examples are given to explain that a soft Ti-I may not to be a soft Ti-II space (i = 0, 1, ⋯ , 4).
This book entitled ‘Recent Development and Techniques in Physical
Sciences’ is presented as a guideline to help researchers to get acquainted with
the new inventions and theories in the field of Physical Sciences. Physical
Science is a collective term for areas of study which includes Physics,
Chemistry, Computer Science, Mathematics, Statistics, Meteorology,
Geology, Astronomy etc. It is a systematic enterprise that build and organize
the acquired knowledge from experiments, observations or predictions; put
them together and try to find their explanation from the laws and formulae. In
this age of science, it is essential and need of time to be aware of the work
going on in different disciplines of science and technology.
The objective of this book is to investigate the recent trends and
developments in science and to keep one’s knowledge abreast with the present
interdisciplinary scenario. The aim is to create interest and understanding of
research and development and to encourage academicians to cope up with the
emerging challenges in their fields in a better and coordinated way.
The present book contains sixteen chapters written by experienced and
eminent scholars, researchers and academicians. The chapters highlight the
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language and are arranged in a very coordinated and logical manner.
We hope that the present form of book will bring awareness about the
variety of interdisciplinary topics. We are thankful to Weser publications,
Germany for their support.
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 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.
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.
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.
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].
The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.
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.
Worldwide, there has been a rapid growth in interest in rough set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on rough sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. In addition, many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs. Rough set theory has led to many interesting applications and extensions. It seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, knowledge discovery, decision analysis, and expert systems. In the article, we present the basic concepts of rough set theory and point out some rough set-based research directions and applications.
After the introduction of fuzzy sets by Zadeh, there have been it number of generalizations of this fundamental concept. The notion Of intuitionistic fuzzy sets introduced by Atanassov is one among them. In this paper, we apply the concept of an intuitionistic fuzzy set to H-v-modules. The notion of an intuitionistic fuzzy H-v-submodule of an H-v-module is introduced, and some related properties are investigated. Characterizations of intuitionistic fuzzy H-v-submodules are given. (c) 2004 Elsevier Ine. All rights reserved.
In this article, we present some extensions of the rough set approach and we outline a challenge for the rough set based research.
In this paper, we apply the theory of soft sets to solve a decision making problem using rough mathematics.
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.
In recent years we witness a rapid growth of interest in rough set theory and its applications, worldwide. The theory has been followed by the development of several software systems that implement rough set operations, in particular for solving knowledge discovery and data mining tasks. Rough sets are applied in domains, such as, for instance, medicine, finance, telecommunication, vibration analysis, conflict resolution, intelligent agents, pattern recognition, control theory, signal analysis, process industry, marketing, etc.
We introduce basic notions and discuss methodologies for analyzing data and surveys some applications. In particular we present applications of rough set methods for feature selection, feature extraction, discovery of patterns and their applications for decomposition of large data tables as well as the relationship of rough sets with association rules. Boolean reasoning is crucial for all the discussed methods.
We also present an overview of some extensions of the classical rough set approach. Among them is rough mereology developed as a tool for synthesis of objects satisfying a given specification in a satisfactory degree. Applications of rough mereology in such areas like granular computing, spatial reasoning and data mining in distributed environment are outlined.
Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (θ,ψ)-fuzzy implicative filters where θ, ψ are any two of {∈,q,∈∨q,∈∧q} with θ≠∈∧q is introduced, and related properties are discussed. Relations between (∈∨q,∈∨q)-fuzzy implicative filters and (∈,∈∨q)-fuzzy implicative filters are investigated, and conditions for an (∈,∈∨q)-fuzzy implicative filter to be an (∈,∈)-fuzzy implicative filter are provided. Characterizations of (∈,∈∨q)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a (q,∈∨q)-fuzzy implicative filter are provided.
The concept of -fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe
the relationships among ordinary fuzzy interior ideals, (鈭� 鈭埪犫埁聽q)-fuzzy interior ideals and -fuzzy interior ideals of semigroups. Finally, we give some characterization of [F]
t
by means of (鈭� 鈭埪犫埁聽q)-fuzzy interior ideals.
Molodtsov introduced the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper we introduce the basic properties of soft sets, and compare soft sets to the related concepts of fuzzy sets and rough sets. We then give a definition of soft groups, and derive their basic properties using Molodtsov’s definition of the soft sets.
The notion of intuitionistic nil radicals of intuitionistic fuzzy ideals in rings is introduced, and related properties are investigated. The notion of semiprime intuitionistic fuzzy ideals is provided, and its characterization is established. The concept of Euclidean intuitionistic fuzzy ideals is also introduced, and its characterization is established.
It is a deep-seated tradition in science to view uncertainty as a province of probability theory. The generalized theory of uncertainty (GTU), which is outlined in this paper, breaks with this tradition and views uncertainty in a broader perspective. Uncertainty is an attribute of information. A fundamental premise of GTU is that information, whatever its form, may be represented as what is called a generalized constraint. The concept of a generalized constraint is the centerpiece of GTU.
The past two decades have witnessed profound changes in the composition, functions and the level of complexity of electrical as well as electronic systems which are employed in modem technology. As a result, classical RLC network theory, which was the mainstay of electrical engineering at a time when RLC networks were the bread and butter of the electrical engineer, has been and is being increasingly relegated to the status of a specialized branch of a much broader discipline-system theory-which is concerned with systems of all types regardless of their physical identity and purpose. This paper presents a brief survey of the evolution of system theory, together with an exposition of some of its main concepts, techniques and problems. The discussion is centered on the notion of state and emphasizes the role played by state-space techniques. The paper concludes with a brief statement of some of the key problems of system theory.
A&gbreve; Man: Soft sets and soft groups
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