Soft Intersection Bi-interior Ideals of Semigroups

Emerging trends in soft set theory and related topics.

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.

Soft rough fuzzy sets and soft fuzzy rough sets

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.

After the paper of Molodtsov [Molodtsov D. Soft set theory — First results, Computers and Mathematics with Applications, 1999, vol. 37, no. 4-5, pp. 19-31.] first appeared, soft set theory grew at a breakneck pace. Several authors have introduced various operations, relations, results, etc. as well as other aspects in soft set theory and hybrid structures incorrectly, despite their widespread use in mathematics and allied areas. In his paper [Molodtsov D.A. Equivalence and correct operations for soft sets, International Robotics and Automation Journal, 2018, vol. 4, no. 1, pp. 18-21.], Molodtsov, the father of soft set theory, pointed out several wrong results and notions. Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] also stated that the concept of soft set had not been fully understood and used everywhere. As a result, it is important to revisit the quirks of those conceptions and provide a formal account of the notion of soft set equivalency. Molodtsov already explored many correct operations on soft sets. We use some notions and results of Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] to create matrix representations as well as related operations of soft sets, and to quantify the similarity between two soft sets.

The concept of soft set is fundamentally important in almost every scientific field. Soft set theory is a new mathematical tool for dealing with uncertainties and is a set associated with parameters and has been applied in several directions. Since Molodtsov originated the idea of soft sets, some research on soft sets has been done in the literature. This theory represents a promising technique in imperfect data analysis which has found interesting extensions and various applications that handle imperfect knowledge, such as Bayesian inference, fuzzy set etc. In this paper, define the notion of soft sets, and the study that are interesting and valuable in the theory of soft sets, which emphasis on a series of applications especially in decision making problems. Also presents comprehensive study, development and survey of its existing literature. Keywords—BCI; BCK; FCM; Fuzzy Set; Fuzzy Soft Set; Soft Rough Set; Soft Semi Rings; Soft Set; Uncertainty.

Soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. This paper is devoted to the further generalization of the soft rough set theory by using the ideal notion to reduce the soft boundary region. A new soft rough set model, called soft ideal approximation space, is proposed and its properties are derived. The new soft ideal lower and upper approximation operators are presented and their related properties are surveyed.

The classical soft rough set theory offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is noticeable that the soft rough set theory cannot be applied in the situation of multigranulation in the real world. To address this issue, in this paper we firstly present two types of multigranulation soft rough sets, which are the optimistic multigranulation soft rough set and the pessimistic multigranulation soft rough set. Then the basic and important properties of these multigranulation soft rough sets are studied. Finally, we investigate the relationships among single granulation soft rough set, namely classical soft rough set, optimistic multigranulation soft rough set and pessimistic multigranulation soft rough set. These results will enrich the soft set theory and multigranulation rough set theory, and also extend their application scopes.

On some new operations in soft set theory

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

Extending fuzzy soft sets with fuzzy description logics

Fuzzy set theory, rough set theory and soft set theory are all generic mathematical tools for dealing with uncertainties. There has been some progress concerning practical applications of these theories, especially, the use of these theories in decision making problems. In the present article, we review some decision making methods based on (fuzzy) soft sets, rough soft sets and soft rough sets. In particular, we provide several novel algorithms in decision making problems by combining these kinds of hybrid models. It may be served as a foundation for developing more complicated soft set models in decision making.

Soft set theory constitutes a mathematically rigorous and algebraically expressive framework for modeling systems characterized by uncertainty, vagueness, and parameter-dependent variability. At the heart of this formalism lies a comprehensive repertoire of algebraic operations and binary product constructions that collectively furnish the universe of soft sets with a rich and finely articulated internal algebraic topology. Within this algebraic landscape, we introduce and formally investigate a novel binary operation, termed the soft union–gamma product, defined over soft sets whose parameter domains are endowed with group-theoretic structure. The operation is rigorously constructed within an axiomatic framework that ensures compatibility with generalized notions of soft subsethood and soft equality. A systematic algebraic analysis is undertaken to establish the operation’s fundamental properties, including closure, associativity, commutativity, idempotency, and distributivity over other soft set operations, as well as its behavior in relation to identity and absorbing elements. Moreover, the product’s interaction with both the null and absolute soft sets is rigorously characterized. Two principal contributions emerge from this investigation. First, the integration of the soft union–gamma product enhances the internal operational harmony of soft set theory by embedding it within a formally consistent, axiomatically governed algebraic environment. Second, the operation lays a conceptual and structural foundation for the development of a generalized soft group theory, wherein soft sets indexed by group-structured parameter domains emulate the axiomatic behavior of classical group-theoretic systems through suitably defined soft operations. This study constitutes a substantive advancement in the algebraic consolidation and theoretical generalization of soft set theory.

As a mathematically consistent and algebraically potent formal system, soft set theory is well-suited for capturing the inherent uncertainty, vagueness, and parameter-based variability in complex decision-making and information systems. Within this theoretical framework, the present study introduces the soft symmetric difference complement–gamma product, a novel binary operation defined on soft sets whose parameter spaces are intrinsically structured by group-theoretic properties. Formulated through a rigorous axiomatic foundation, the operation is shown to maintain full compatibility with generalized soft subsethood and soft equality, thereby ensuring its structural integration within the broader algebraic fabric of soft set theory. A comprehensive algebraic examination is conducted to identify and verify the operation’s key structural properties, including closure, associativity, commutativity, and idempotency. The operation’s interactions with identity and absorbing elements, as well as with the null and absolute soft sets, are rigorously analyzed within the constraints imposed by group-parameterized domains. The findings confirm that the soft symmetric difference complement–gamma product satisfies all required algebraic conditions dictated by group-theoretic parameter structures, resulting in a coherent and robust algebraic framework over the universe of soft sets. In addition to its foundational value, the operation substantially enriches the structural landscape of soft set theory and offers a formal pathway toward the construction of a generalized soft group theory, where soft sets indexed by group-based parameters emulate classical algebraic operations. Its consistency with generalized notions of soft equality and its seamless integration into layered soft inclusion hierarchies further emphasize its algebraic depth and applicability. Accordingly, this work not only delivers a significant algebraic innovation but also provides a principled platform for extending soft set theory to fields that demand formal mechanisms for handling uncertainty, abstract algebraic modeling, and multi-criteria decision-making frameworks.

Interval-valued intuitionistic fuzzy soft sets and their properties