I am Professor Aslıhan Sezgin, a faculty member in the field of Pure Mathematics, with a specialization in Abstract Algebra. My research interests primarily lie in soft set theory and algebraic structures, particularly soft groups and soft symmetric difference groups.
My work focuses on the development of new theoretical frameworks, rigorous algebraic definitions, and the investigation of structural properties within group theory. I have contributed to academic publications in international journals and have been actively involved in research projects and scientific events in pure mathematics.
My academic goal is to advance the field of algebra through original research and to collaborate with researchers working in abstract algebra and related mathematical disciplines.
Soft Set Theory; Soft Groups; Group-Theoretic Structures; Abstract Algebra
In this paper, we define soft intersection semigroups, soft intersection left (right, two-sided) ideals and bi-ideals of semigroups, give their properties and interrelations and we characterize regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups in terms of these ideals.
This study aims to reveal the bibliometric characteristics of publications on educational research from diverse perspectives, including the level of national-international collaborations, the percentage change in open-access papers, and interactions with other disciplines. Through bibliometric analysis, the data were collected from Web of Science (WoS), “Education and Educational Research (E&ER),”“Psychology, Educational (PE),”“Education, Special (ES)” and “Education Scientific Disciplines (ESD)” categories, during 2011 to 2020. The findings suggest that the number of publications in each category and the percentage of open-access publications have increased regularly. However, it was observed that categories, especially the ES category, were partially falling behind, as compared to the growth in WoS; while the ES and the PE categories were the highest regarding national-international collaborations over time. Both collaboration levels were the highest in the E&ER category. It was concluded that the multi-authorship trend has rapidly increased, the number of references has regularly increased, and specifically, the PE category has been distinguished from others in terms of the references used. Meanwhile, the change over time in the aggregate impact factor, calculated for each category as a measure of the widespread impact of the studies, pointed to inflation. The results show that educational research is becoming more organized and international, while less fragmented. Moreover, the findings support that educational research is interdisciplinary in essence and diverse in content.
The soft set theory developed by Molodtsov has been applied both theoretically and practically in many fields. It is a useful piece of mathematics for handling uncertainty. Numerous variations of soft set operations have been described and used since its introduction. In this paper, we define a new soft set operation, called complementary soft binary piecewise intersection operation, a unique soft set operation, and examine its basic algebraic properties. Additionally, we aim to contribute to the literature on soft sets by illuminating the relationships between this new soft set operation and other kinds of soft set operations by researching the distribution of this new soft set operation over other soft set operations. Moreover, we prove that the set of all the soft sets with a fixed parameter set together with the complementary soft binary piecewise intersection operation and the soft binary piecewise union operation is a zero-symmetric near-semiring and hemiring.
Soft set theory, introduced by Molodtsov, is as an important mathematical tool to deal with uncertainty and it has been conveyed to many fields both as theoretical and application aspect. Since its inception, different kinds of soft set operations are defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise star operation and we investigate its basic algebraic properties. Moreover, it is aimed to contribute to the soft set literature by obtaining the relationships between this new soft set operation and some other types of soft set operations by examing the distribution of complementary soft binary piecewise star operation over extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations
Molodtsov’s soft set theory has been used in various disciplines both theoretically and practically. It is an effective mathematical tool for dealing with uncertainty. Since its debut, numerous types of soft set operations have been presented and applied. In this study, we analyze the fundamental algebraic features of a novel soft set operation which we call complementary soft binary piecewise union operation. Additionally, by examining the distribution of complementary soft binary piecewise union operation over all other soft set operations’ type, it aims to contribute to the literature on soft sets by revealing relationships between this new soft set operation and other types of soft set operations. Moreover, we prove that the set of all the soft sets with a fixed parameter set together with the complementary soft binary piecewise union operation and the soft binary piecewise intersection operation is a zero-symmetric near-semiring.
Molodtsov initiated the soft set theory, providing a general mathematical framework for handling uncertainties that we encounter in various real-life problems. The main object of this paper is to lay a foundation for providing a new soft algebraic tool for considering many problems that contain uncertainties. In this paper, we introduce a new kind of soft ring structure called [Formula: see text]-soft-intersectional ring based on some results of soft sets and intersection operations on sets. We also define [Formula: see text]-soft-intersectional ideal and [Formula: see text]-soft-intersectional subring, and investigate some of their properties using these new concepts. We obtain some results in ring theory based on [Formula: see text]-soft intersection sense and its application in ring structures. Furthermore, we provide relationships between soft-intersectional ring and [Formula: see text]-soft-intersectional ring, soft-intersectional ideal and [Formula: see text]-soft-intersectional ideal.
Soft set theory is a theory of dealing with uncertainty. Since its inception, many kinds of soft set operations are defined and used in various types. In this paper, a new kind of soft set operation called, complementary soft binary piecewise difference operation is defined and its basic properties are investigated. We obtain many striking analogous fact between difference operation in classical theory and complementary soft binary piecewise difference operation in soft set theory. Also, by obtaining the relationships between this new soft set operation and all other types of soft set operations, we aim to contribute to the soft set literature with the help of examing the distribution rules.
Set theory is considered as the foundation of all mathematics since many mathematical concepts cannot be defined precisely without using set-theoretical concepts. In this study, we define new complemental binary operations, called union complements, intersection left complement, and union right complement and investigate their properties in detail. We contribute to the literature of sets by illuminating the relationships between these complemental binary operations and inclusive\exclusive complements via researching the distribution rules. Moreover, we show that the set of all the sets together with these new complemental binary operations form some algebraic structures. Finally, with the inspiration of these novel concepts, we give an application to group theory as regards subgroups by defining new type of subgroups in order to prompt the reader to think via interesting questions. Since the concept of operations of soft set theory, one of the most popular theory for uncertainty modeling in the past twenty four years, is the crucial notion for developing the theory and since all the types of soft set operations are based on the classical set operations, generation of new complemental binary operations on sets, and thus on soft sets and derivation of their algebraic properties will provide new perspectives for solving problems related to parametric data.
Soft set theory, developed by Molodtsov, has been applied both theoretically and practically in many fields. It is a useful mathematical tool for handling uncertainty. Numerous variations of soft set operations, which is a crucial concept for the theory, have been described and used since its introduction. In this paper, we explore more about soft binary piecewise difference operation (defined first as “difference of soft sets”) and its whole properties are examined especially in comparison with the basic properties of difference operation in classical set theory. Several striking properties of soft binary piecewise operations are obtained as analogous to the characteristic of difference operation in classical set theory. Also, we show that the collection of all soft sets with a fixed parameter set together with the soft binary piecewise difference operation is a bounded BCK-algebra.
Cubic sets are the very useful generalization of fuzzy sets where one is allowed to extend the output through a subinterval of [ 0 , 1 ] and a number from [ 0 , 1 ] . Generalized cubic sets generalized the cubic sets with the help of cubic point. On the other hand Soft sets were proved to be very effective tool for handling imprecision. Semigroups are the associative structures have many applications in the theory of Automata. In this paper we blend the idea of cubic sets, generalized cubic sets and semigroups with the soft sets in order to develop a generalized approach namely generalized cubic soft sets in semigroups. As the ideal theory play a fundamental role in algebraic structures through this we can make a quotient structures. So we apply the idea of neutrosophic cubic soft sets in a very particular class of semigroups namely weakly regular semigroups and characterize it through different types of ideals. By using generalized cubic soft sets we define different types of generalized cubic soft ideals in semigroups through three different ways. We discuss a relationship between the generalized cubic soft ideals and characteristic functions and cubic level sets after providing some basic operations. We discuss two different lattice structures in semigroups and show that in the case when a semigroup is regular both structures coincides with each other. We characterize right weakly regular semigroups using different types of generalized cubic soft ideals. In this characterization we use some classical results as without them we cannot prove the inter relationship between a weakly regular semigroups and generalized cubic soft ideals. This generalization leads us to a new research direction in algebraic structures and in decision making theory.
Molodtsov introduced the theory of soft sets, which can be seen as an effective mathematical tool to deal with uncertainties, since it is free from the difficulties that the usual theoretical approaches have troubled. In this paper, we apply the definitions proposed by Ali et al. [M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009), 1547-1553] to the concept of soft near- rings and substructures of soft near-rings, proposed by Atag?n and Sezgin [A. O. Atag?n and A. Sezgin, Soft Near-rings, submitted] and show them with illustrating examples. Moreover, we investigate the properties of idealistic soft near-rings with respect to the near-ring mappings and we show that the structure is preserved under the near-ring epimorphisms. Main purpose of this paper is to extend the study of soft near-rings from a theoretical aspect.
The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, we introduce the concepts of soft union hemirings (soft union h-ideals) of hemirings by soft intersection-union product and obtain some related results. Finally, we investigate some characterizations of h-hemiregular hemirings by soft union h-ideals.
In this paper, soft intersection LA-semigroups, soft intersection left (right, two-sided) ideals, (generalized) bi-ideals, interior ideals, quasi-ideals in LA-semigroups are defined, their examples and properties are given and their interrelations are obtained.
Soft set theory, initiated by Molodtsov, is a tool for modeling various types of uncertainty. In this paper, upper and lower α-inclusions of a soft set are defined. By using these new notions, some analyzes with respect to group theory are made and it is shown that some of the subgroups of a group can be obtained easily with the help of these notions. It is also illustrated that a soft int-group and a soft uni-group can be obtained by its upper α-subgroups and lower α-subgroups, respectively. Furthermore, soft int-group by its family of upper α-subgroups is characterized under a certain equivalence relation. Finally, a new method used to construct a soft int-group with the help of its upper α-subgroups are introduced and an application of this method is given.
Abstract In this work, soft covered ideals in semigroups are constructed and in this concept, soft covered semigroups, soft covered left (right) ideals, soft covered interior ideals, soft covered (generalized) bi-ideals and soft covered quasi ideals of a semigroup are defined. Various properties of these ideals are introduced and the interrelations of these soft covered ideals and the relations of soft anti covered ideals and soft covered ideals are investigated.
In this paper, we define soft union near-ring on a soft set by using soft sets, inclusion relation and union of sets. This new notion functions as a bridge among soft set theory, set theory and near-ring theory. We then derive its basic properties and investigate the relationship between soft intersection near-ring and soft union near-ring. Furthermore, we obtain some analog of classical near-ring theoretic concepts for soft union near-ring and give the applications of soft union near-ring to near-ring theory.
Soft set theory, introduced by Molodtsov, is as an efficacious mathematical tool to deal with uncertainty and it has been applied to many fields both as theoretical and application aspect. Since its inception, different kinds of soft set operations are defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise gamma operation and we investigate its basic algebraic properties. Moreover by examing the distribution rules, it is aimed to contribute to the soft set literature by obtaining the relationships between this new soft set operation and other types of soft set operations such as extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations. This paper can be regarded as a theoritical study of soft set theory.
In this paper, a new approach to LA-semigroup theory is proposed by obtaining significant characterizations of regular, intra-regular, completely regular, weakly regular and quasi-regular LA-semigroups via soft intersection left (right, two-sided) ideals, (generalized) bi-ideals, interior ideals, quasi-ideals of LA-semigroups.
In this paper, the concepts soft union AG-groupoids, soft union left (right, two-sided) ideals, (generalized) bi-ideals, interior and quasi-ideals in AG-groupoids are introduced with many illustrating examples and their properties and interrelations are given. Moreover, regular, intra-regular, completely regular, weakly regular and quasi-regular AG-groupoids are characterized by the properties of these soft union ideals.
Molodtsov tarafından geliştirilen esnek küme teorisi hem teorik hem de pratik olarak birçok alanda uygulanmıştır. Belirsizliği ele almak için yararlı bir matematiksel araçtır. Ortaya atıldığından bu yana çok sayıda esnek küme işlemi varyasyonu tanımlanmış ve kullanılmıştır. Bu çalışmada, esnek ikili parçalı simetrik fark işlemi adı verilen yeni bir esnek küme işlemi tanımlanıp, özellikleri klasik küme teorisinde var olan simetrik fark işleminin temel cebirsel özellikleri ile karşılaştırmalı olarak ele alınmış ve incelenmiştir. Ayrıca, esnek ikili parçalı simetrik fark işlemi ve kısıtlanmıs kesisim işlemleri ile birlikte sabit parametreye sahip tüm esnek kümelerin oluşturduğu cebirsel yapının, birimli ve değişmeli bir hemiring ve ayrıca Boole halkası olduğu gösterilmiştir.
In this paper, soft union ring (SU-ring) on a soft set is defined by using union operation of sets.This new concept shows how a soft set effects on a ring structure in the mean of union and inclusion of sets and from this overview, it functions as a bridge among soft set theory, set theory and ring theory.Then, its basic properties are derived and the relationship between soft intersection ring defined in [N.C ¸agman and F. C ¸ıtak and H. Aktas ¸, Soft intrings and its algebraic applications, Journal of Intelligent and Fuzzy Systems, 28 (3): 1225-1233(2015)] and SU-ring are investigated.Furthermore, we give the applications of SU-ring to ring theory.
In 1999, Molodtsov introduced Soft Set Theory as a mathematical tool to deal with uncertainty. It has been applied to many fields both as theoretical and application aspects. Since 1999, different kinds of soft set operations have been defined and used in various types. In this paper, we define a new kind of soft set operation called, “complementary soft binary piecewise lambda operation” and we handle its basic algebraic properties. Also, it is intended to contribute to the literature of soft set by gaining the relationships between this new soft set operation and some other types of soft set operations via examining the distribution of complementary soft binary piecewise lambda operation over extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations in order to inspire to obtain the algebaric structures of soft sets and some new decision making methods.
In this paper, assuming that N is a near-ring and P is an ideal of N, the P-center of N, the P-center of an element in N, the P-identities of N are defined. Their properties and relations are investigated. It is shown that the set of all P-identities in N is a multiplicative subsemigroup of N. Also, P-right and P-left permutable and P-medial near-rings are defined and some properties and connections are given. P-regular and P-strongly regular near-rings are studied. P-completely prime ideals are introduced and some characterizations of P-completely prime near-rings are provided. Also, some properties of P-idempotents, P-centers, P-identities in P-completely prime near-rings are investigated. The results that were obtained in this study are illustrated with many examples.
Abstract 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 semilattice 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.
Soft Set Theory, which has been considered as an adequate mathematical device, was proposed by Molodtsov to deal with ambiguities and uncertainties. Several operations on soft sets were defined in many soft set papers. This study is based on the paper "On operations of Soft Sets" by Sezgin and Atag\"{u}n [Comput. Math. Appl. 61 (2011) 1457-1467]. In this paper, we define a new operation on soft sets, called extended difference and investigate its relationship between extended difference and restricted difference and some other operations of soft sets.
Soft set theory, pioneered by Molodtsov in 1999, presents a soft framework for managing uncertainty in data analysis and decision-making. In contrast to conventional set theory, soft sets permit elements to possess parametrization, offering a more intricate portrayal of uncertainty. In this paper, we introduce a novel type of soft set operation known as complementary extended theta soft set operations to contribute the existing theory. We thoroughly analyze the properties of this operation and investigate the relationship between the complementary extended theta operation and other soft set operations in order to further study of algebraic structures of soft sets with respect to the new operation in the future studies.
Since its beginnings, soft set theory has shown to be a useful mathematical framework for addressing problems involving uncertainty, proving its usefulness in a variety of academic and practical disciplines. The operations of soft sets are at the very core concept of this theory. In this regard, a new kind of soft set operation known as the complementary extended gamma operation for soft sets is presented in order to improve the theory and theoretically contribute to it in this study. To shed light on the relation between the complementary extended gamma operation and other soft set operations, a thorough analysis of this operation's attributes, including its distributions across other soft set operations, has been conducted. Additionally, this paper aims to contribute to the literature on soft sets by examining the algebraic structure of soft sets from the perspective of soft set operations, which provides a thorough grasp of their use as well as an appreciation of the ways in which soft sets can be applied to both classical and nonclassical logical thought.
In this paper, we aim to extend the studies [M. R. Alimoradi, R. Rezaei and M. Rahimi, Some notes on ideals in soft rings, Journal Australian Journal of Basic and Applied Sciences 6(3) (2012) 717–721; F. Koyuncu and B. Tanay, Some soft algebraic structures, Journal of New Results Science 10(2016) 38–51] as regards maximal, prime and principal soft ideal of soft rings, characterize soft rings with these soft ideals and also provide some more relations between maximal, prime and principal soft ideals of soft rings. The notions of maximality and primeness points of soft ideals of a soft rings are defined, maximal and prime idealistic soft rings as well as maximal, prime and principal soft ideal of soft rings and their basic properties are more investigated under certain conditions, especially by means of homomorphism and epimorphism of rings. We apply some of the basic results about maximal ideals and prime ideals in classical abstract algebra to maximal, prime and principal idealistic soft rings and we give some of their interrelations between each others.
This paper aims to introduce the novel concept of neutrosophic crisp soft set (NCSS), including various types of neutrosophic crisp soft sets (NCSSs) and their fundamental operations. We define NCS-mapping and its inverse NCS-mapping between two NCS-classes. We develop a robust mathematical modeling with the help of NCS-mapping to analyze the emerging trends in social networking systems (SNSs) for our various generations. We investigate the advantages, disadvantages, and natural aspects of SNSs for five generations. With the changing of the generations, it is analyzed that emerging trends and the benefits of SNSs are increasing day by day. The suggested modeling with NCS-mapping is applicable in solving various decision-making problems.
Semigroups are the building blocks of algebra as they have application in automata, coding the-ory, formal languages, and theoretical computer science. They are also used in the solutions of graph theory and optimization theory. For the advanced study of algebraic structures and their applications, ideals are essential. The generalization of ideals in algebraic structures is necessary for further research on algebraic structures. The main purpose of this paper is to present the notion of soft intersection almost subsemigroup of a semigroup, which is a generalization of soft intersection subsemigroup and investigate its basic pro-perties in detail. In this context, we also obtain many striking relationships between almost subsemigroups and soft intersection almost subsemigroups concerning minimality, primeness, semiprimeness and strongly primeness.
The notions of left (right) ideal and quasi-ideal of semigroup are generalized by the concept of tri-ideal. Likewise, the concepts of soft intersection left (right) ideal and soft intersection quasi-ideal of semigroups are generalized by the soft intersection tri-ideal. In this paper, we present and study the concept of soft intersection almost tri-ideal ideal as a further generalization of nonnull soft intersection tri-ideal. It is demonstrated that every idempotent soft intersection almost bi-ideal is a soft intersection almost tri-ideal, and vice versa. We also obtained that an idempotent soft intersection almost left (or right) tri-ideal coincides with the soft intersection almost tri-ideal. It is also shown that every idempotent soft intersection almost tri-ideal is a soft intersection almost subsemigroup. With the noteworthy result that if a nonempty subset of a semigroup is an almost tri-ideal, then its soft characteristic function is also a soft intersection almost tri-ideal, and vice versa, a number of intriguing relationships regarding minimality, primeness, semiprimeness, and strongly primeness between almost tri-ideals and soft intersection almost tri-ideals are derived.
The aim of this study is to present the notion of soft intersection almost left (respectively, right) ideal of a semigroup which is a generalization of nonnull soft intersection left (respectively, right) ideal of a semigroup and investigate the related properties in detail. We show that every idempotent soft intersection almost (left/right) ideal is a soft intersection almost subsemigroup. Besides, we acquire remarkable relationships between almost left (respectively, right) ideals and soft intersection almost left (respectively, right) ideals of a semigroup as regards minimality, primeness, semiprimeness and strongly primeness.
<p>Soft sets provide a strong mathematical foundation for managing uncertainty and inventing solutions to parametric data problems. Soft set operations are fundamental elements within soft set theory. In this paper, we introduce a new product operation for soft sets, called the “soft lambda-product,” and thoroughly examine its algebraic properties in relation to various types of soft equalities and subsets. By studying the distribution of the soft lambda-product over different soft set operations, we further investigate its relationship with other soft set operations. We conclude with an example demonstrating the method’s effectiveness across various applications, employing the <em>int-uni</em> operator and <em>int-uni</em> decision function within the soft lambda-product for the <em>int-uni</em> decision-making method, which identifies an optimal set of elements from available options. This work significantly contributes to the soft set literature, as the theoretical foundations of soft computing methods rely on solid mathematical principles.</p>
Soft sets provide a comprehensive mathematical framework for tackling uncertainty. Soft set operations and products are fundamental to soft set theory, offering innovative solutions to problems that involve parametric data. In this study, we first adapted the soft L-subsets/equality concept and soft J-subsets/equality for the revised soft set concept. Additionally, we defined some new types of soft subsets and equalities, called soft S-subsets/equality and soft A-subsets, along with their specific examples to clarify these concepts. We investigated the connections among these new concepts. This paper presents an innovative product for soft sets whose parameter sets are groups, called the "soft symmetric difference-difference-product". We thoroughly analyzed its fundamental algebraic properties, considering various soft subsets and equality relations to inspire future research. This may lead to a new soft group theory arising from this concept. Since soft algebraic structure theories are grounded in soft set operations and products, this study contributes significantly to the literature on soft sets.
After being presented by Molodtsov in 1999, soft set theory became well-known as a novel strategy for resolving uncertainty-related issues and modeling uncertainty. It has several uses in both theoretical and real-world settings. In this study, a novel soft set operation known as the "soft binary piecewise theta operation" is presented. Its fundamental algebraic properties are investigated in detail. Furthermore, the distributions of this operation over other soft set operations are examined. In addition to being a right-left system under certain circumstances, we demonstrate that the soft binary piecewise theta operation is also a commutative semigroup in the collection of soft sets over the universe. Furthermore, by taking into account the algebraic properties of the operation and its distribution rules together, we demonstrate that the collection of soft sets over the universe, along with the soft binary piecewise theta operation and some other types of soft sets, form many important algebraic structures, like semirings and nearsemirings.
For handling uncertainty, the theory of soft sets provides a thorough mathematical foundation. Soft set operations are significant concepts in soft set theory, as they offer new approaches to problems involving parametric data. In this context, we introduce a new product operation for soft sets, called “soft theta-product” and investigate its whole algebraic properties in terms of different types of soft subsets and soft equalities. Additionally, we explore the relations of this soft product with other soft set operations by investigating the distributions of the soft theta-product over them. In conclusion, using the uni-int decision function for the soft-theta product together with the uni-int operator for the uni-int decision-making method, which chooses a collection of optimal elements from the alternatives, we provide an example demonstrating how the technique may be effectively applied in a range of areas. As the theoretical underpinnings of soft computing techniques are drawn from purely mathematical concepts, this study is an essential contribution to the literature on soft sets.
Soft sets provide a strong mathematical foundation for managing uncertainty and give creative answers to parametric data challenges. In soft set theory, soft set operations are essential components. The “soft gamma-product,” a novel product operation for soft sets, is presented in this study along with a detailed analysis of its algebraic features with respect to different kinds of soft equalities and subsets. We further explore the soft gamma-product’s relation with other soft set operations by examining its distributions over other soft set activities. Using the uni-int operator and uni-int decision function within the soft gamma-product for the uni-int decision-making approach, which finds an ideal collection of components from accessible possibilities, we end with an example showing the method's efficacy of many applications. Since the theoretical underpinnings of soft computing techniques are based on sound mathematical concepts, this study makes a substantial contribution to the literature on soft sets.
In this paper, soft union semigroups, soft union left (right, two-sided) ideals and bi-ideals of semigroups are defined, their properties and interrelations are given and regular, intra-regular, completely regular, weakly regular and quasi-regular semigroups are characterized in terms of these ideals. This paper is a new approach to classical semigroup theory via soft set theory.
Keeping in view the expediency of soft sets in algebraic structures and as a mathematical approach to vagueness, in this paper the concept of lattice ordered soft near rings is introduced. Different properties of lattice ordered soft near rings by using some operations of soft sets are investigated. The concept of idealistic soft near rings with respect to lattice ordered soft near ring homomorphisms is deliberated.
The aim of this study is to identify the most prolific countries in the field of special education and to discuss the widespread impact of their papers by taking into account the country’s h-index. Through a bibliometric analysis, the data were collected in the Web of ScienceCore Collection category “Education, Special” in the Social Science Citation Index during 2011-2020. The 25 most prolific countries in the field of special education were determined in terms of paper productivity, and it was seen that the leading country was undisputedly the USA (54.42%). Meanwhile, a strong positive correlation was found between the h-index and the number of papers published by the countries (r=0.864). On the other hand, when the ranking in terms of the number of papers was reconfigured by the h-index, it was relatively changed. The possible reasons for this change for the countries with the most changing rankings were discussed by considering some definitive criteria such as the journal quartiles, the percentage of international and domestic, and the percentage of open access papers.This study reports a positive correlation between the quality and quantity in the field of special education for the publications of countries. It has been shown that where the positive correlation deviates, then especially, the journal quartiles, the percentage of international collaboration and the percentage of open access papers have a significant effect. The bibliometric findings may be useful to enrich the discussion about the widespread impact of papers and debate whether the use of h-index is acceptable for cross-national comparisons.
Similar to how the quasi-interior ideal generalizes the ideal and interior ideal of a semigroup, the concept of soft intersection quasi-interior ideal generalizes the idea of soft intersection ideal and soft intersection interior ideal of a semigroup. In this study, we provide the notion of soft intersection almost quasi-interior ideal as well as the soft intersection weakly almost quasi-interior ideal in a semigroup. We show that any nonnull soft intersection quasi-interior ideal is a soft intersection almost quasi-interior ideal; and soft intersection almost quasi-interior ideal is a soft intersection weakly almost quasi-interior ideal, but the converses are not true. We further demonstrate that any idempotent soft intersection almost quasi-interior ideal is a soft intersection almost subsemigroup. With the established theorem that states that if a nonempty set A is almost quasi-interior ideal, then its soft characteristic function is a soft intersection almost quasi-interior ideal, and vice versa, we are also able to derive several intriguing relationships concerning minimality, primeness, semiprimeness, and strongly primeness between almost quasi-interior ideals, and soft intersection almost quasi-interior ideals.
Just as the concept of interior ideal of semigroups is a generalization of ideal in semigroups, the notion of soft intersection (soft-int) interior ideal is a generalization of soft-int ideal. In this paper, we propose the concepts of soft-int (weakly) almost interior ideal of a semigroup as a generalization of the nonnull soft-int interior ideals. We explore their algebraic properties in detail. We also show that an idempotent soft-int almost interior ideal is a soft-int almost subsemigroup. We additionally derive several intriguing relations related to semiprimeness, minimality, and (strongly) primeness between almost interior ideals and soft-int almost interior ideals.
This study aims to introduce the concept of soft intersection almost weak-interior ideals of a semigroup, which extends the notion of nonnull soft intersection weak-interior ideals of a semigroup. We explore the properties of the ideal in depth. We show that soft intersection almost ideal and soft intersection almost weak-interior ideal coincide with each other when the soft set is idempotent and we also illustrate that an idempotent soft almost weak-interior ideal is a soft intersection almost subsemigroup. We also establish significant connections between almost weak-interior ideals and soft intersection weak-interior ideals of a semigroup concerning minimality, primeness, semiprimeness, and strong primeness
Soft set theory gained popularity as a cutting-edge approach to handling uncertainty-related problems and modeling uncertainty when it was introduced by Molodtsov in 1999. It may be applied in a variety of contexts, both theoretical and practical. This paper introduces a new soft set operation called the “soft binary piecewise star operation.” Its basic algebraic characteristics are thoroughly examined. Moreover, this operation’s distributions over various soft set operations are obtained. We prove that the soft binary piecewise star operation is a commutative semigroup under certain conditions and is also a right-left system. Furthermore, we show that the collection of soft sets over the universe, along with the soft binary piecewise star operation and some other types of soft sets, form many important algebraic structures, such as semirings and near-semirings, by considering the algebraic properties of the operation and its distribution rules together.
Objective: Bone marrow involvement (BMI) is a significant component of staging of Hodgkin's lymphoma (HL). Unilateral bone marrow biopsy (BMB) from dorsal iliac bone is the standard method to determine BMI. Positron emission tomography/computerized tomography (PET/CT) is recommended as a complimentary technique to determine BMI and to evaluate response to treatment. The aim of this study is to determine whether PET/CT can replace BMB to detect BMI in patients with HL. Methods: A total of 159 patients diagnosed as having HL were evaluated retrospectively. One hundred and four patients who met the criteria were included in the study. BMB and PET/CT were performed on all patients during initial staging. Results: Of the 104 patients, 44 (42.3%) and 17 (16.3%) had BMI in PET/CT and BMB respectively. All patients who had BMI in BMB also had involvement in PET/CT. BMB did not detect BMI in 27 patients who had BMI in PET/CT. All 27 patients had partial or complete remission on PET/CT performed at the end of treatment. This finding was regarded as a relative indicator for BMI. PET/CT had a sensitivity and negative predictive value of 100%, specificity of 68.9% and accuracy rate of 74% to determine BMI. Conclusion: We observed that PET/CT and BMB were compatible to determine BMI in patients diagnosed with HL. Unilateral BMB may result false negative especially in cases with focal involvement. Therefore, PET/CT should be considered as a complimentary technique to determine BMI. Staging should be reevaluated in patients who have BMI in PET/CT and a negative BMB treatment should be planned accordingly.
,Soft set theory gained popularity as a cutting-edge approach to handling uncertainty-related problems and modeling uncertainty after being introduced by Molodtsov in 1999. Numerous theoretical and practical applications have been conducting by make using of the theory. This paper presents a novel soft set operation, called the &quot;soft binary piecewise gamma operation&quot;. Its basic algebraic characteristics are thoroughly examined. Furthermore, this operation&#039;s distributions over a number of soft set operations are investigated. We prove that given certain assumptions, the soft binary piecewise addition operation determines a commutative semigroup and a right-left system, and soft binary piecewise gamma operation, along with the some other types of soft sets, form many important algebraic structures, such as semirings and near-semirings in the collection of soft sets over the universe by considering the algebraic properties of the operation and its distribution rules together.
Since its introduction by Molodtsov in 1999, soft set theory has gained widespread recognition as a method for addressing uncertainty-related issues and modeling uncertainty. It has been used to solve several theoretical and practical issues. Since its introduction, the central idea of the theory-soft set operations-has captured the attention of scholars. Numerous limited and expanded businesses have been identified, and their attributes have been scrutinized thus far. We present a detailed analysis of the fundamental algebraic properties of our proposed restricted theta and extended theta operations, which are unique restricted and extended soft set operations. We also investigate these operations' distributions over various kinds of soft set operations. We demonstrate that, when coupled with other types of soft set operations, the extended theta operation forms numerous significant algebraic structures, such as semirings in the collection of soft sets over the universe, by taking into account the algebraic properties of the extended theta operation and its distribution rules. This theoretical subject is very important from both a theoretical and practical perspective since soft sets' operations form the foundation for numerous applications, including cryptology and decision-making procedures.
Critical thinking, a term referring to a broad range of cognitive skills and intellectual tendencies necessary for effectively identifying, analyzing, and evaluating arguments, is an essential tool for solving any problem. It is especially crucial in solving mathematical problems. Specifically, in this paper, using critical thinking as well as fundamental knowledge of mathematical logic, we highlight certain shortcomings, in terms of precision, in some approaches to solving irrational inequalities.
A thorough mathematical foundation for handling uncertainty is provided by the concept of soft sets. Soft set operations are key concepts in soft set theory since they offer novel approaches to problems requiring parametric data. The “soft difference-product” a new product operation for soft sets, is proposed in this study along with all of its algebraic properties concerning different types of soft equalities and subsets. Additionally, we explore the connections between this product and other soft set operations by investigating the distributions of soft difference-product over other soft set operations. Using the uni-int operator and the uni-int decision function for the soft-difference product, we apply the uni-int decision-making method, which selects a set of optimal elements from the alternatives by giving an example that shows how the approach may be conducted effectively in various areas. Since the theoretical underpinnings of soft computing techniques are drawn from purely mathematical concepts, this study is crucial to the literature on soft sets.
Soft set theory, introduced by Molodtsov in 1999, offers a versatile framework for handling uncertainty in data analysis and decision-making processes. Unlike traditional set theory, soft sets allow elements to be parametrized, providing a more nuanced representation of uncertainty. In this context, a new kind of soft set operation called complementary extended plus soft set operation, is defined in this paper to contribute to the theory. The properties of the operation are examined in detail, along with its distributions over other soft set operations, to establish the relationship between the complementary extended plus operation and other operations.
It has been shown that generalizing the ideals of an algebraic structure is both interesting and beneficial for mathematicians. In this context, the concept of quasi-interior (Ԛꟾ) ideal was introduced as a generalization of quasi-ideal and interior ideal of a semigroup. In this paper, we apply this concept to soft set theory and semigroups, introducing a new form of soft intersection (S-int) ideal called the "soft intersection (S-int) quasi-interior (Ԛꟾ) ideal." The main objective of this study is to investigate the relationships between S-int Ԛꟾ ideals and other specific types of S-int ideals in a semigroup. It has been shown that every S-int interior ideal of a semigroup is an S-int Ԛꟾ ideal, and every S-int ideal is an S-int Ԛꟾ ideal. The S-int bi-ideal of a group is an S-int Ԛꟾ ideal, the S-int quasi-ideal of a regular group is an S-int Ԛꟾ ideal, the idempotent S-int Ԛꟾ ideal is an S-int bi-quasi-ideal and an S-int bi-interior ideal. Counterexamples are provided to show that the opposites of these statements are not always valid. We prove that for the converses to hold, the semigroup should be a group or regular, or the S-int Ԛꟾ ideal should be idempotent. Our main theorem, which demonstrates that if a subsemigroup of a semigroup is a Ԛꟾ ideal, then its soft characteristic function is an S-int Ԛꟾ ideal, and vice versa, enables us to establish a connection between semigroup theory and soft set theory. Through this theorem, we illustrate how this concept connects to the existing algebraic structures in classical semigroup theory. Additionally, we offer conceptual characterizations and an analysis of the concept in terms of soft set operations, including soft image and soft inverse image, supporting our claims with specific, informative examples. Furthermore, the connection between a regular semigroup and the structure of S-int Ԛꟾ ideals is established and presented.
Mathematicians find it valuable to extend the concept of ideals within algebraic structures. The bi-quasi (ƁԚ) ideal was introduced as a broader version of quasi-ideal, bi-ideal, and left (right) ideals in semigroups. This paper applies this concept to soft set theory and semigroups, introducing the "Soft intersection (S-int) ƁԚ ideal." The goal is to explore the relationships between S-int ƁԚ ideals and other types of S-int ideals in semigroups. It is shown that every S-int bi-ideal, S-int ideal, S-int quasi-ideal, and S-int interior ideal of an idempotent soft set are S-int ƁԚ ideals. Counterexamples demonstrate that the reverse is not always true unless the semigroup is simple* or regular. For soft simple* semigroups, the S-int ƁԚ ideal coincides with the S-int bi-ideal, S-int left (right) ideal, and S-int quasi-ideal. The main theorem shows that if a subsemigroup of a semigroup is a ƁԚ ideal, its soft characteristic function is an S-int ƁԚ ideal, and vice versa. This connects semigroup theory with soft set theory. The paper also discusses how this concept integrates into classical semigroup structures, providing characterizations and analysis using soft set operations, soft image, and soft inverse image, supported by examples.
Soft set theory has gained prominence as a revolutionary approach for handling uncertainty-related problems and modeling uncertainty since it was proposed by Molodtsov. The concept of soft set operations, which is the major notion for the theory, has served as the foundation for theoretical and practical advances in the theory, therefore deriving the algebraic properties of the soft set operations and studying the algebraic structure of soft sets associated with soft set operations have attracted the researchers’ interest continuously. In the theory of soft set, many soft intersection operations have been defined up to now among which there are some differences, and some of which are no longer preferred for use as they are essentially not useful and functional. Although the definition of restricted intersection is widely accepted in the literature and used in the studies, it is still incomplete with its current form suffering from certain cases where the parameter sets of the soft sets may be disjoint is ignored, thus all the circumstances in the theorems are not considered in the related proofs causing to the incorrectness or deficiency in the studies where this operation is used or its properties are investigated. In this regard, in the existing literature, there is a critical lack of comprehensive study on the correct defined restricted intersection operation together with extended intersection including their correct properties and distributions and the correct algebraic structures assoiciated with these soft set operations. In this study, we primarly intend to fill this crucial gap by first correcting the deficiencies in the presentation of the definition of restiricted intersection and revising it. Moreover, in many papers related to these operation, several theorems were presented without their proofs, or there were some incorrect parts in the proofs. In this study, all the proofs based on the function-equality are regularly provided and besides, the relationships between the concept of soft subset and restricted and extended intersection operations are presented for the first time with their detailed proofs. Furhermore, we obtain many new properties of these operations as analogy and counterpart of intersection operation in classical set theory. Moreover, the operations’ full properties and distributions over other soft set operations are throughly investigated to determine the correct algebraic structures the operations form individually and in combination with other soft set operations both in the set of soft sets over the universe and with a fixed parameter set. We demonstrate that the restricted/extended intersection operations, when combined with other kinds of soft set operations, form several significant algebraic structures, such as monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, Kleene Algebra, Stone algebra and MV-algebra but with deteailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted intersection and extended intersection in the literature to date as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by filling the significant gap in the literature, acting as a guide for the beginners of this popular theory, and besides shedding light on the future studies on soft sets.
A rigorous and expressive algebraic framework for modeling systems with uncertainty, ambiguity, and parameter-dependent variability is offered by soft set theory. In this paper, we present a new binary operation on soft sets whose parameter domains have group-theoretic structure: the soft union–plus product. The operation is completely compatible with generalized concepts of soft equality and soft subsethood when it is specified formally inside an axiomatic framework. Key structural aspects such as closure, associativity, commutativity, idempotency, and distributivity are investigated in detail algebraically, as is its behavior with respect to identity, absorbing, null, and absolute soft sets. The outcomes demonstrate that the operation creates a strong and cohesive algebraic system on the universe of soft sets while adhering to all algebraic limitations imposed by group-indexed domains. In addition to its theoretical importance, the operation provides a strong basis for a generalized soft group theory and reinforces the underlying algebraic architecture of soft set theory. Furthermore, it has significant potential for both abstract theoretical advancement and real-world applications due to its formal consistency with soft subset and equality relations, which improves its usefulness in domains like categorization, decision-making, and uncertainty-aware modeling.
Soft set theory constitutes a highly flexible and mathematically rigorous framework for modeling and analyzing real-world phenomena characterized by uncertainty, ambiguity, and parameter-dependent variability—features that frequently arise in disciplines such as decision sciences, engineering, economics, and information systems. Central to this theoretical apparatus are the fundamental operations and product constructions on soft sets, which collectively give rise to a rich and expressive algebraic infrastructure capable of accommodating complex parametric interdependencies. In this study, we introduce a novel product, termed the soft union–difference product, specifically defined for soft sets whose parameter sets possess a group structure. A thorough axiomatic and structural analysis of this is conducted, with special attention to its algebraic compatibility with generalized notions of soft subsethood and soft equality. Through this analysis, we uncover the product’s intrinsic structural properties and demonstrate its capacity to preserve essential algebraic features within group-parameterized soft set systems. Furthermore, we conduct a comprehensive algebraic investigation of the soft union–difference product, examining its closure, associativity, idempotency, commutativity, absorbing property, and distributivity, as well as its interaction with other established soft products defined on groups and null soft sets. These investigations reveal two pivotal theoretical implications: first, they reinforce the internal algebraic coherence of soft set theory by situating the newly defined product within a formally consistent operational framework; second, they lay a conceptual foundation for the emergence of a soft group theory that structurally parallels classical group-theoretic constructions. Given that the advancement of soft algebraic systems is inherently predicated on rigorously defined operations and systematically articulated product frameworks, the present study makes a substantial contribution to the formal algebraic refinement and theoretical evolution of soft set theory. Beyond their theoretical merit, the proposed constructions also offer concrete methodological tools for the development of group-based soft computational models, with potential applications in multi-criteria decision-making, uncertainty-aware classification systems, and data-driven analysis under parameter uncertainty.
Since its introduction by Molodtsov in 1999, soft set theory has gained significance as an innovative approach for handling uncertainty-related issues and modeling uncertainty. It has several applications in both theoretical and real-world settings. Researchers have been interested in soft set operations, the theory's fundamental concept, since its inception. Several restricted and extended soft set operations, one of which is restricted and extended difference operation, were defined, and their characteristics were partially investigated. The present study gives a full analysis of the properties of restricted and extended difference soft set operations, particularly in comparison to the fundamental characteristics of the difference operation existing in classical set theory, as the existing studies regarding restricted and extended set operations are quite incomplete in that all the properties of the operations have been examined in depth. In this regard, this paper is a complete study of the aforesaid soft set operations. We also observe several noteworthy properties that bear analogies to the difference operation of classical set theory. Furthermore, by investigating the distributions of these operations over all other types soft set operations including the new types of soft set operations, and taking into account the algebraic properties of the operation and its distribution rules, we demonstrate that the extended difference operation, when combined with other kinds of soft set operations, forms several significant algebraic structures, such as semiring and near-semirings in the collection of soft sets over the universe, This theoretical study has considerable significance both theoretically and practically, as the fundamental concept of the soft set theory is the operations of soft sets since they serve as the foundation for numerous applications, including cryptology, as well as the decision-making processes as correctly as possible.
Since its introduction by Molodtsov, soft set theory has developed in prominence as an innovative method for dealing with uncertainty-related problems and modeling uncertainty. Soft set operations, the theory's main concept, have served as the foundation for theoretical and practical advances in the theory; thus, deriving the algebraic properties of soft set operations and studying the algebraic structure of soft sets associated with soft set operations has piqued researchers' interest continuously. Many soft union operations have been proposed in soft set theory, but there are some differences. Even though the definition of restricted union is widely acknowledged in the literature and applied in many studies, it is still lacking in its current form owing to the fact that a specific case where the soft sets' parameter sets may be disjoint is ignored in the definition. As a result, all the cases in the theorems are not taken into account, leading to errors or deficiencies in the studies that use this operation or investigate this operation’s properties. Regarding this, a thorough examination of the properly defined restricted union operation and extended union operation, together with their appropriate distributions and properties, as well as the appropriate algebraic structures connected to these soft set operations, is conspicuously lacking in the body of current literature. This study is primarily intended to fill this critical and significant gap by first fixing the presentational flaws of the restricted union definition and updating it. Furthermore, in many works on these operations, numerous theorems were offered without their proofs, or the proofs had some incorrect parts. In this paper, all the proofs based on function equality are supplied regularly, and the relations between the concept of soft subset and restricted and extended union operations are obtained for the first time with their extensive proofs. Furthermore, we explore numerous novel properties of these operations as analogies and counterparts of the union operation in classical set theory. We show that when restricted/extended union operations are combined with other types of soft set operations, several significant algebraic structures are formed, including monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, and MV-algebra with detailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted union and extended union in the literature to date, as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by bridging a significant gap in the literature, serving as a guide for newcomers to this popular theory, and providing insight for further future research.
Soft set theory provides a mathematically rigorous and algebraically expressive framework for modeling systems characterized by epistemic uncertainty, vagueness, and parameter-dependent variability—phenomena central to decision theory, engineering, economics, and information science. Expanding on this foundation, the present study introduces and examines a novel binary operation, the soft intersection–star product, defined over soft sets with parameter domains possessing intrinsic group-theoretic structures. Developed within a formally consistent, axiomatic framework, this operation aligns with generalized concepts of soft subsethood and soft equality. A comprehensive algebraic analysis is on the operation’s core properties—closure, associativity, commutativity, and idempotency. The presence or absence of identity, inverse, and absorbing elements, and the soft product’s behavior concerning the null and absolute soft sets, are precisely delineated. To contextualize the operation, a comparative analysis with prior binary soft products is conducted, elucidating its expressive capacity and structural coherence within the layered hierarchy of soft subset classifications. The findings demonstrate that the soft in-tersection–star product satisfies all axiomatic requirements imposed by group-parameterized domains, thereby inducing a robust and internally consistent algebraic structure on the space of soft sets. Two key contributions emerge: first, the operation substantially extends the algebraic toolkit of soft set theory within a rigorous opera-tional framework; second, it lays the foundation for a generalized soft group theory, wherein soft sets indexed by group-structured parameters mimic classical group behavior through abstractly defined soft operations. Beyond its theoretical value, the proposed framework offers a principled basis for soft computational modeling grounded in abstract algebra. Such models are highly applicable to multi-criteria decision analysis, algebraic classification, and uncertainty-sensitive data analytics. Hence, this study not only strengthens the theoretical foundations of soft algebra but also reinforces its relevance to both mathematical research and practical computation.
In this paper, we introduce union soft subrings and union soft ideals of a ring and union soft submodules of a left module and investigate their related properties with respect to so ft set operations, anti image and lower α-inclusion of soft sets. We also obtain significant relation between soft subrings and union soft su brings, soft ideals and union soft ideals, soft submodules a nd union soft submodules.
In this paper, soft union interior ideals, quasi-ideals and generalized bi-ideals of rings are defined and their properties are obtained and the interrelations of them are given. Moreover regular, regular duo, intra-regular and strongly regular rings are characterized in terms of these soft union ideals. This paper is a following study of [19].
Since Molodtsov first introduced soft set theory, a useful mathematical tool for solving problems related to uncertainties, many soft set operations have been described and used in decision making problems. In this study, a new soft set operation called complementary soft binary piecewise symmetric difference operation is defined, and its properties are examined in comparison with the basic algebraic properties of the symmetric difference operation. Moreover, it has been shown that the collection of soft sets with a fixed set of parameter together with complementary soft binary symmetric difference and restricted intersection, is a commutative hemiring with identity and also a Boolean ring.
Soft set theory was proposed by Molodtsov in 1999 to model some problems involving uncertainty. It has a wide range of theoretical and practical applications. Soft set operations constitute the basic building blocks of soft set theory. Many kinds of soft set operations have been described and applied in various ways since the inception of the theory. In this paper, to contribute to the theory, a new soft set operation, called complementary extended union operation, is defined, its properties are discussed in detail to obtain the relationship of each operation with other soft set operations, and the distributions of these operations over other soft set operations are examined. We obtain that the complementary extended union operation along with other certain types of soft set operations construct some well-known algebraic structure such as Boolean Algebra, De Morgan Algebra, semiring, and hemiring in the set of soft sets with a fixed parameter set. Since Boolean Algebra is fundamental in digital logic design, computer science, information retrieval, set theory and probability; De Morgan Algebra in logic and set theory, computer science, artificial intelligence, circuit design; semirings in theoretical computer science, optimization problems, economics, cryptography and coding theory, and hemirings in combinatorics, mathematical economics, theoretical computer science, these algebraic structures provide essential tools for various applications, facilitating the analysis, design, and optimization of systems across many disciplines, and thus this study is expected to contribute to decision-making methods and cryptography based on soft sets.
In settings where parametric variability is present, soft set theory has evolved into a robust and versatile mathematical framework for modeling and analyzing uncertainty. Central to this framework are the operations and product constructions on soft sets, which together provide a powerful algebraic infrastructure for addressing complex parameter-dependent problems. Formally, it is shown that under the union operation, the collection of all soft sets defined over a fixed parameter set forms a bounded semilattice, thereby supplying essential algebraic structure and coherence. Through careful analysis, it is further demonstrated that the algebraic system consisting of all soft sets over a fixed parameter set endowed with a group structure, equipped with the union operation and the proposed product, satisfies the axiomatic framework of a hemiring. This structural characterization yields two major theoretical implications: it strengthens the algebraic foundations of soft set theory and lays the groundwork for constructing a soft group theory analogous to its classical counterpart. A new product on soft sets, termed the soft symmetric difference–union product, is then introduced for the case in which the parameter set carries a group structure. This operation is examined in detail from both axiomatic and structural perspectives, with particular attention to its compatibility with soft equality and soft subsethood. The approach presented here makes a substantial contribution to the ongoing algebraic refinement and theoretical advancement of soft set theory, as the formal development of soft algebraic systems fundamentally relies on rigorously defined operations and product constructions.
In this paper, we define the concepts of soft anti-covered semigroup, soft anti-covered left (right) ideal, soft anti-covered interior ideal, soft anti-covered (generalized) bi-ideal, soft anti-covered quasi-ideal of a semigroup. We investigate their properties by various examples. Moreover, we focus on the interrelations of them. We construct a new approach to anti-covered ideals of a semigroup via soft set theory.
Molodtsov, in 1999, introduced soft set theory as a mathematical tool to deal with uncertainty. Since then, different kinds of soft set operations have been defined and used in various types. In this paper, it is aimed to contribute to the soft set literature by obtaining the distributions of soft binary piecewise operations over complementary soft binary piecewise plus and gamma operations.
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 semigroup.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.
In order to deal with uncertainty, Molodtsov propoed soft set theory as a mathematical tool in 1999. Since that time, numerous forms of soft set operations have been defined and employed. By establishing the soft binary piecewise operations&#039; distributions over complementary soft binary piecewise intersection and union operations, this study aims to contribute to the literature on soft set theory by giving inspiration to researchers as regards examining and obtaining some algebraic structures using these new soft set operations.
1999 yılında Molodtsov tarafından ortaya atılan esnek küme teorisi, belirsizlikle başa çıkmak için bir araçtır. O zamandan beri, farklı türden esnek küme işlemleri tanımlanmış ve çeşitli türlerde kullanılmıştır. Bu çalışmada esnek ikili parçalı işlemlerin tümleyenli esnek ikili parçalı fark ve lambda işlemleri üzerinden dağılımları elde edilerek esnek küme literatürüne katkı sağlanması amaçlanmaktadır.
Soft set theory has established itself as a valuable mathematical framework for tackling issues marked by uncertainty, demonstrating its applicability across a range of theoretical and practical fields since its inception. Central of this theory is the operations of soft sets. To enhance the theory and to make a theoretical contribution to the theory, a new type of soft set operation, called “complementary extended star operation” for soft set, is proposed. An exhaustive examination of the properties of this operation has been undertaken, including its distributions over other soft set operations, with the goal of clarifying the relationship between the complementary extended star operation and other soft set operations. This paper also attempts to make a contribution to the literature of soft sets in the sense that studying the algebraic structure of soft sets from the standpoint of soft set operations offers a comprehensive understanding of their application as well as an appreciation of how soft set can be applied to classical and nonclassical logic.
Soft set theory is seen as an effective mathematical tool in solving problems involving uncertainty, and has been applied in many theoretical and practical areas since its introduction. The basic concept of the theory is soft set operations. In this context, in this paper, a new kind of soft set operations called complementary extended soft set operation is defined in order to contribute to the theory. The properties of the operation are examined in detail together with its distributions over other soft set operations in order to obtain the relationship between complementary extended intersection operation and the others. We demonstrate that the collection of soft sets over with a fixed parameter set, along with the complementary extended intersection operation and other certain types of soft sets, form many well-known and important algebraic structures in classical algebra, including semiring, hemiring, Boolean ring, Boolean Algebra, De Morgan Algebra, Kleene Algebra, and Stone Algebra.
This paper aims to expand soft int-group theory by analyzing its many aspects and structural properties regarding soft cosets and soft quotient groups, which are crucial concepts of the theory. All the characteristics of soft cosets are given in accordance with the properties of classical cosets in abstract algebra, and many interesting analogous results are obtained. It is proved that if an element is in the e-set, then its soft left and right cosets are the same and equal to the soft set itself. The main and remarkable contribution of this paper to the theory is that the relation between the e-set and the normality of the soft int-group is obtained, and it is proved that if the e-set has an element other than the identity of the group, then the soft int-group is normal. Based on this significant fact, it is revealed that if the soft set is not normal, then there do not exist any equal soft left (right) cosets. These relations are quite striking for the theory, since based on these facts, we show that the normality condition on the soft int-group is unnecessary in many definitions, propositions, and theorems given before. Furthermore, we come up with a fascinating result, unlike classical algebra that to construct a soft quotient group and to hold the fundamental homomorphism theorem, the soft int-group needs not to be normal. It is also demonstrated that the soft int-group is an abelian (normal) int-group if and only if the soft quotient group of G relative to the soft group is abelian. Finally, the torsion soft-int group and 𝑝-soft int-group are introduced, and we show that soft int-group f_G is a torsion soft-int group (𝑝-soft int-group) if and only if the soft quotient group G⁄f_G is a torsion (𝑝-group), respectively.
Soft set theory has many theoretical and practical applications. It was first introduced by Molodtsov in 1999 as a way to represent specific situations including uncertainty. The fundamental building blocks of soft set theory are soft set operations. Since its debut, several types of soft set operations have been defined and utilized in diverse contexts. In order to further the theory, a new soft set operation known as the complementary extended difference operation is defined in this paper. Its properties are thoroughly discussed, with particular attention to how it differs from the difference operation in classical sets. Additionally, the distribution of this operation over other types of soft set operations is examined in order to determine how this operation relates to other soft set operations.
In classical algebra, p-groups, conjugate groups and Sylow Theorems are of great importance to understand the arbitrary finite group structures. Our interest, in this paper, is to transfer these important structures to soft group theory. First, we define soft prime group, conjugate group and soft conjugate group. Then, we examine their properties under group mappings, group homomorphisms and soft homomorphisms. Also, as a strong case of conjugate group, strong conjugate group is defined and the relationship between the conjugation and strong conjugation is derived and it is showed that strong conjugation is an equivalence relation on the set of all soft groups over G with the parameter set A. Additionally, we convey Cauchy’s Theorem to soft groups. Moreover, in order to understand the structure of an arbitrary finite soft group, we define soft Sylow p-subgroup and obtain the corresponding Sylow Theorems in soft group theory with this concept. By this way, we bring a new aspect to soft group theory by expanding the theory with the fundamental concepts.
Mathematicians attach importance to extending ideals in algebraic structures. The concept of bi-quasi (ƁԚ) ideal was introduced as a generalization version of quasi-ideal, bi-ideal, and left (right) ideals in semigroups. This paper applies this concept to soft set theory and semigroups, introducing the notion of "Soft union (S-uni) ƁԚ ideal." The aim of this paper is to explore the relationships between S-uni ƁԚ ideals and other types of S-uni ideals in semigroups. It is shown that every S-uni bi-ideal, S-uni ideal, S-uni quasi-ideal, and S-uni interior ideal of an idempotent soft set are S-uni ƁԚ ideals. Counterexamples demonstrate that the converses are not always true unless the semigroup is special soft simple or regular. For special soft simple semigroups, the S-uni ƁԚ ideal coincides with the S-uni bi-ideal, S-uni left (right) ideal, and S-uni quasi-ideal. Additionally, we provide conceptual definitions and analyses of the new concept in the context of soft set operations, supporting our claims with clear examples.
Generalizing the ideals of an algebraic structure has shown to be both beneficial and interesting for mathematicians. In this context, the idea of the bi-interior ideal was introduced as a generalization of the bi-ideal and interior ideal of a semigroup. By introducing "soft union (S-uni) bi-interior ideals of semigroups", we apply this idea to semigroups and soft set theory in this study. Finding the relationships between S-uni bi-interior ideals and other specific kinds of S-uni ideals of a semigroup is the main aim of this study. Our results show that an S-uni bi-interior ideal is an S-uni subsemigroup of a special soft simple semigroup, and that the S-uni bi-interior ideal of semigroup is a generalization of the S-uni left (right/two-sided) ideal, bi-ideal, interior ideal, and quasi-ideal, however, the converses are not true with counterexamples. We demonstrate that the semigroup should be a special soft simple semigroup in order to satisfy the converses. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of regarding soft set operations and notions supporting our assertions with particular, illuminating examples.
The concept of the soft union (S-uni) bi-quasi-interior ₿ꝖĪ) ideal of semigroups is proposed in this study, along with its equivalent definition. We derive the relationships between S-uni ideals and S-uni ₿ꝖĪ ideal. The S-uni ₿ꝖĪ ideal is shown to be S-uni bi-ideal, left ideal, right ideal, interior ideal, quasi-ideal, bi-interior ideal, left/right bi-quasi ideal, and left/right quasi-interior ideal. It is shown that certain additional requirements, such as regularity or right/left simplicity, are necessary for the converses, and counterexamples are given to demonstrate that the converses are not true. Additionally, it is demonstrated that the soft anti characteristic function of a subsemigroup of a semigroup is an S-uni ₿ꝖĪ ideal if the subsemigroup itself is a ₿ꝖĪ ideal, and vice versa. Consequently, a significant connection between soft set theory and classical semigroup theory is established. Additionally, it is demonstrated that while the finite soft OR-products and union of S-uni ₿ꝖĪ ideals are also S-uni ₿ꝖĪ ideals, the intersection and finite soft AND-products are not. A broad conceptual characterization and analysis of S-uni ₿ꝖĪ ideals are presented in this paper.
Since its introduction by Molodtsov in 1999, soft set theory has gained significance as an innovative approach for handling uncertainty-related issues and modeling uncertainty. It has several applications in both theoretical and real-world settings. Researchers have been interested in soft set operations, the theory's fundamental concept, since its inception. Several restricted and extended soft set operations, one of which is restricted and extended difference operation, were defined, and their characteristics were partially investigated. The present study gives a full analysis of the properties of restricted and extended difference soft set operations, particularly in comparison to the fundamental characteristics of the difference operation existing in classical set theory, as the existing studies regarding restricted and extended set operations are quite incomplete in that all the properties of the operations have been examined in depth. In this regard, this paper is a complete study of the aforesaid soft set operations. We also observe several noteworthy properties that bear analogies to the difference operation of classical set theory. Furthermore, by investigating the distributions of these operations over all other types soft set operations including the new types of soft set operations, and taking into account the algebraic properties of the operation and its distribution rules, we demonstrate that the extended difference operation, when combined with other kinds of soft set operations, forms several significant algebraic structures, such as semiring and near-semirings in the collection of soft sets over the universe, This theoretical study has considerable significance both theoretically and practically, as the fundamental concept of the soft set theory is the operations of soft sets since they serve as the foundation for numerous applications, including cryptology, as well as the decision-making processes as correctly as possible.
Soft set theory constitutes a mathematically rigorous and algebraically versatile framework for modeling systems characterized by epistemic uncertainty, vagueness, and parameter-dependent variability. Building upon this foundational structure, the present study introduces and thoroughly investigates a novel binary operation, termed the soft symmetric difference-plus product, defined on soft sets whose parameter set is a group. This operation is axiomatized within a logically coherent and formally consistent framework, ensuring full compatibility with generalized notions of soft subsethood and soft equality. A comprehensive algebraic analysis is undertaken to establish the fundamental properties of the operation, including closure, associativity, commutativity, and idempotency. Moreover, the existence or absence of the identity and the absorbing elements of the product, along with its characteristics relative to the null and absolute soft sets, are explicitly characterized. This inquiry elucidates the relative expressive capacity, algebraic coherence, and structural integrability of the soft symmetric difference-plus product within the layered hierarchies of soft subset classifications. The results demonstrate that the operation satisfies all necessary axiomatic requirements dictated by group-parameterized domains, thereby inducing a robust and internally consistent algebraic structure on the universe of soft sets. Two principal contributions emerge: first, the introduction of the soft symmetric difference-plus product substantially enriches the operational repertoire of soft set theory by embedding it within a rigorously defined, operation-preserving algebraic framework; second, it lays a conceptual foundation for the advancement of a generalized soft group theory, wherein soft sets indexed by group-structured parameter domains emulate classical group behaviors through abstractly formulated soft operations. Beyond its theoretical significance, the framework developed herein provides a mathematically principled basis for constructing soft computational models grounded in abstract algebra. Such models hold considerable promise for applications in multi-criteria decision-making, algebraic classification, and uncertainty-aware data analysis.
Soft set theory, as a mathematically rigorous and algebraically expressive formalism, offers a powerful framework for modeling uncertainty, vagueness, and parameter-driven variability. Within this landscape, the present study introduces the soft symmetric difference complement-intersection product, a novel binary operation defined over soft sets whose parameter domains are endowed with a group-theoretic structure. Developed within a strict axiomatic foundation, the operation is proven to satisfy fundamental algebraic properties—such as closure, associativity, commutativity, and idempotency—while maintaining consistency with generalized notions of soft equality and subsethood. Its behavior is thoroughly analyzed with respect to identity and absorbing elements, as well as interactions with null and absolute soft sets, all within the constraints of group-parameterized domains. The findings confirm that the proposed operation forms a coherent and structurally robust algebraic system, thereby enriching the algebraic architecture of soft set theory. Furthermore, this work provides a foundational step toward the formulation of a generalized soft group theory, in which soft sets indexed by group-based parameters emulate classical group behaviors through abstract soft operations. The operation’s full integrability within soft inclusion hierarchies and its alignment with generalized soft equalities highlight its theoretical depth and broaden its potential applications in formal decision-making and algebraic modeling under uncertainty.
Generalizing the ideals of an algebraic structure has shown to be both beneficial and interesting for mathematicians. In this context, the idea of the bi-interior ideal was introduced as a generalization of the bi-ideal and interior ideal of a semigroup. By introducing "soft intersection (₷-int) bi-interior (ᙝᏆ) ideals of semigroups", we introduce a framework integrating semigroup theory with soft set theory in this study. Finding the relationships between ₷-int ᙝᏆ-ideals and other specific kinds of ₷-int ideals of a semigroup is the main aim of this study. Our results show that an ₷-int ᙝᏆ-ideal is an ₷-int subsemigroup of a soft simple* semigroup, and that an ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal is an ₷-int ᙝᏆ-ideal; in other words, the ₷-int ᙝᏆ-ideal is a generalization of the ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal, however, we provide counterexamples demonstrating that the converses do not always hold. We demonstrate that the semigroup should be a soft simple* semigroup in order to satisfy the converses. Our key theorem, which states that if a nonempty subset of a semigroup is a ᙝᏆ-ideal, then its soft characteristic function is an ₷-int ᙝᏆ-ideal, and vice versa, enables us to bridge the gap between semigroup theory and soft set theory. Using this theorem, we show how this idea relates to the existing algebraic structures in classical semigroup theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of soft set operations supporting our assertions with illuminating examples.
Soft set theory provides a logically sound and algebraically rich framework for modeling systems characterized by ambiguity, epistemic uncertainty, and parameter-dependent variability. This study introduces the soft union–lambda product, a novel binary operation defined over soft sets whose parameter spaces are endowed with an intrinsic group-theoretic structure. Developed within a rigorously formulated axiomatic framework, the opera-tion is shown to be fully compatible with generalized notions of soft subsethood and soft equality. A thorough algebraic investigation is undertaken to establish the fundamental structural properties of the opera-tion—including closure, associativity, commutativity, idempotency, and its distributivity over other soft set operations—alongside a precise characterization of its behavior with respect to identity, absorbing, null, and absolute soft sets. The results demonstrate that the soft union–lambda product adheres to all algebraic con-straints dictated by group-parameterized domains, thereby inducing a consistent and internally cohesive alge-braic structure on the universe of soft sets. Beyond its foundational contributions, the proposed operation sig-nificantly expands the formal toolkit of soft set theory and sets the stage for the development of a generalized soft group theory. Furthermore, its formal alignment with key relational structures such as soft equality and soft inclusion highlights its potential applicability across a diverse array of analytical contexts, including abstract algebraic modeling, uncertainty-aware classification, and multi-criteria decision analysis. Accordingly, the findings of this study offer both profound theoretical advancements and concrete pathways for practical im-plementation.
This paper provides a theoretical investigation of the OR-product (∨-product) in soft set theory, an operation of central importance for handling uncertainty in decision-making. A comprehensive algebraic analysis is carried out with respect to various types of subsets and equalities, with particular emphasis on M-subset and M-equality, which represent the strictest forms of subsethood and equality. This framework reveals intrinsic algebraic symmetries, particularly in commutativity, associativity, and idempotency, which enrich the structural understanding of soft set theory. In addition, certain missing results on OR-products in the literature are completed, and our findings are systematically compared with existing ones, ensuring a more rigorous theoretical framework. A central contribution of this study is the demonstration that the collection of all soft sets over a universe, equipped with a restricted/extended intersection and the OR-product, forms a commutative hemiring with identity under soft L-equality. This structural result situates the OR-product within one of the most fundamental algebraic frameworks, connecting soft set theory with broader areas of algebra. To illustrate its practical relevance, the int-uni decision-making method on the OR-product is applied to a pilot recruitment case, showing how theoretical insights can support fair and transparent multi-criteria decision-making under uncertainty. From an applied perspective, these findings embody a form of symmetry in decision-making, ensuring fairness and balanced evaluation among multiple decision-makers. By bridging abstract algebraic development with concrete decision-making applications, the results affirm the dual significance of the OR-product—strengthening the theoretical framework of soft set theory while also providing a viable methodology for applied decision-making contexts.
Soft set theory has emerged as a novel approach to modelling uncertainty, addressing a wide range of theoretical and practical problems. In this study, we define a new soft set operation called “soft binary piecewise intersection operation” and explore its fundamental algebraic properties by comparing them with the properties of the intersection operation in classical set theory. Many remarkable similarities have been observed between the intersection operation and the soft binary piecewise intersection operation. Additionally, we examine the distribution of the soft binary piecewise intersection operation over other types of soft set operations. By analysing the algebraic properties of the operation and its distribution rules, we demonstrate that the collection of soft sets over the universe, as well as the collection of soft sets with a fixed parameter set, alongside the soft binary piecewise intersection operation and other types of soft sets, forms several important algebraic structures such as hemirings, near-semirings, semirings, Boolean rings, Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras.
The concept of tri-quasi ideal was presented as a generalization of quasi-ideal, interior ideal, and bi-ideal. In this paper, we transfer this concept to soft set theory and semigroups,and introduce a novel type of soft union (S-uni) ideal form called "soft union (S-uni) tri-bi-ideal”. The main aim of this study is to obtain the relations between S-uni tri-bi-ideals and other certain types of S-uni ideals of a semigroup. Our results show that every S-uni tri-bi-ideal of a band is an S-uni subsemigroup. Moreover, an S-uni tri-bi-ideal is a generalization of an S-uni ideal, interior ideal, bi-ideal, quasi-ideal, weak-interior ideal, bi-interior ideal and bi-quasi ideal, however in order to satisfy the converses, the semigroup should have specific conditions. We also demonstrate that the S-uni quasi-interior ideal of a left or right simple semigroup is an S-uni tri-bi-ideal, nevertheless the converse holds for the zero semigroup. Furthermore, the S-uni bi-quasi-interior ideal of a commutative semigroup is an S-uni tri-bi-ideal, however, for the converse to hold the semigroup must be a band. We have shown that an S-uni tri-ideal coincides with an S-uni tri-bi-ideal of a band, and every S-uni tri-bi-ideal of a group is an S-uni tri-ideal. We also obtain a relation between tri-bi-ideal and its soft characteristic function, enabling us to get the relation between semigroup and soft set theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of the soft set operations, the soft (anti/inverse) image, supporting our assertions with illuminating examples.
I am seeking research collaborators working on soft set theory, including its extensions and related mathematical structures. The collabora…