Journal articles on the topic 'Non Null Soft Set'

The concept of soft sets was proposed as an effective tool to deal with uncertainty and vagueness. Topologists employed this concept to define and study soft topological spaces. In this paper, we introduce the concepts of soft SD-continuous, soft SD-open, soft SD-closed and soft SD-homeomorphism maps by using soft somewhere dense and soft cs-dense sets. We characterize them and discuss their main properties with the help of examples. In particular, we investigate under what conditions the restriction of soft SD-continuous, soft SD-open and soft SD-closed maps are respectively soft SD-continuous, soft SD-open and soft SD-closed maps. We logically explain the reasons of adding the null and absolute soft sets to the definitions of soft SD-continuous and soft SD-closed maps, respectively, and removing the null soft set from the definition of a soft SD-open map.

In this paper, we introduce the concept of Possibility Fermatean Neutrosophic Soft Set and define some related concepts such as Possibility Fermatean Neutrosophic Soft subset, Possibility Fermatean Neutrosophic Soft null set, and Possibility Fermatean Neutrosophic Soft universal set. Then, we define set-theoretical operations of Possibility Fermatean Neutrosophic Soft Sets such as union, intersection, and complement, and investigate some properties of these operations. We also introduce AND-product and OR-product operations between two Possibility Fermatean Neutrosophic Soft Sets. We propose a decision-making method called the Possibility Fermatean Neutrosophic Soft decision-making method (PFNS-decision-making method) which can be applied to decision-making problems involving uncertainty based on AND-product operation. We finally give a numerical example to display the application of the method that can be successfully applied to the problems.

We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size [Formula: see text], such that every transversal of this partition is null.

The classical set theory based on crisp sets is not able to deal with uncertainties which is a common feature of various real-world problems. This problem is solved using modified forms of sets such as fuzzy sets, Intuitionistic fuzzy sets, neutrosophic sets, soft sets and hypersoft sets and others along with their hybrids. In this paper, a modified hybrid of soft set named Fermatean Neutrosophic Soft set ($FrNSS$) is established. Basic entities of set theory including subsets, null set, universal set along with different operators are defined. With respect to these operators, the algebraic structures as monoid, semigroup and semiring are defined. Also, fermatean neutrosophic soft topological space and the cartesian product of fermatean neutrosophic soft sets and fermatean neutrosophic soft relation are defined to establish an application of this hybrid structure to decision-making problems.

Bu yazıda iki matematiksel yaklaşımı belirsizliğe göre ilginç bir bağlantı veriyoruz: bipolar yumuşak kümeler ve ikili yumuşak kümeler. İkili iki kutuplu yumuşak darı iki evrensel küme üzerine set ve bir parametre seti önerilmiştir. Tamamlayıcı, birleşme, kesişme, sınırlı birleşme, sınırlı kesişme, null ikili bipolar yumuşak seti, mutlak ikili bipolar yumuşak seti, iki ikili bipolar yumuşak seti, “VE”, “VEYA” tarihinde ikili bipolar yumuşak setlerde tanımlanmıştır. İkili bipolar yumuşak kümelerin temel özellikleri de araştırılmış ve tartışılmıştır. Sonunda ikili bipolar yumuşak kümenin karakteristik bir fonksiyonunu veriyoruz.

A generalized soft set model that is more accurate, useful, and realistic is the bipolar spherical fuzzy soft set (BSFSs). It is a more developed variant of current fuzzy soft set models that may be applied to characterize erroneous data in practical applications. Bipolar spherical fuzzy soft sets and bipolar spherical fuzzy soft topology are novel ideas that are intended to be introduced in this work. Bipolar spherical fuzzy soft intersection, bipolar spherical fuzzy soft null set, spherical fuzzy soft absolute set, and other operations on bipolar spherical fuzzy soft sets are some of the fundamental ideas defined in this work. The bipolar spherical fuzzy soft open set, the bipolar spherical fuzzy soft close set, the bipolar spherical fuzzy soft closure, and the spherical fuzzy soft interior are also defined. Additionally, the characteristics of this specified set are covered and described using pertinent instances. The innovative notion of BSFSs makes it easier to describe the symmetry of two or more objects. Moreover, a group decision-making algorithm based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) approach to problem-solving is described. We analyze the symmetry of the optimal decision and ranking of feasible alternatives. A numerical example is used to show how the suggested approach may be used. The extensive benefits of the proposed work over the existing techniques have been listed.

Molodtsov has introduced the concept of soft sets and the application of soft sets in decision making and medical diagnosis problems. The basic properties of vague soft sets are presented. In this paper, we introduce the concept of interval-valued vague soft sets which are an extension of the soft set and its operations such as equality, subset, intersection, union, AND operation, OR operation, complement, and null while further studying some properties. We give examples for these concepts, and we give a number of applications on interval-valued vague soft sets.

Abstract This paper is about the distribution of null and strong pronouns in French prepositional phrases in relation to the humanness of the preposition’s complement. The existing literature suggests that the mapping between pronoun type and humanness is characterized by soft complementarity, linking human complements to strong pronouns, and non-human complements to null pronouns. In the empirical coverage of the phenomenon experimental studies are still underrepresented: so far, no data on pronoun choice is available, and regarding interpretation data, the only existing study does not include null pronouns. Two experiments were conducted with the goal to fill this research gap. The results show that soft complementarity only applies to pronoun choice: human complements are typically referred to by strong pronouns, and non-human complements by null pronouns. As related to interpretation, however, both strong and null pronouns are preferably interpreted as human. We account for this asymmetry based on the general preference of pronouns to be interpreted as human unless prevented by strong constraints (as in the case of Engl. it).

The objective of this research is to explore ternary soft sets over three initial universal sets, incorporating a parameter set, also known as the set of decision variables. Fundamental operations such as subset, superset, equality, complement, null set, and absolute set are examined, along with the union and intersection of two ternary soft sets. The study further investigates the difference and symmetric difference between two ternary soft sets, as well as the “AND” and “OR” operations, particularly in relation to the crisp points of the sets. The behavior and properties of ternary soft sets are analyzed, with examples provided for clarification. A novel mathematical structure, ternary soft topological structures, is introduced, focusing on three initial universal sets and a fixed parameter set. Key concepts such as ternary soft open sets, closed sets, closures, interiors, boundaries, and neighborhoods are defined and explored in depth. The relationships between these concepts are examined to provide a comprehensive understanding. Illustrative examples demonstrate the practical applications of these ideas. Additionally, ternarysoft semi-separation axioms, along with various properties, such as ternary soft semi-regular,semi-normal, and invariance properties, including the ternary soft topological and hereditary properties, are discussed.

AbstractA subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

Groups and subgroups are rich algebraic structures, and both of them depend on binary operations in their work. The discussion of this paper is organized into two parts. In the first part, we define the notion of Qcomplex neutrosophic soft sets (Q-CNSSs) by amalgamating two previous models of Q-complex neutrosophic set (Q-CNS) and soft set (SS) to address the issues of two-dimensionality (two variables) in a universal set under a parametric environment. Subsequently, the relation between Q-CNSSs and Q- neutrosophic soft sets (Q-NSSs) is verified. A basic set theory for this hybrid model is developed. In particular, null Q-CNSS and absolute Q-CNSS are defined. The basic operators of the complement, subset, equality, union and intersection are advanced and their properties are examined. Further, the notions of the homogeneous and completely homogeneous Q-CNSSs are proposed along with some illustrated examples. In part two, we move to study some algebraic structures of this model when we define the notions of Q-complex neutrosophic soft groups (QCNSG) and Q-complex neutrosophic soft subgroups (Q-CNSSG). Then, the relation between Q-CNSG and Q-neutrosophic soft group (Q-NSG) is scrutinized. Moreover, the algebraic properties of the Q-CNSG and Q-CNSSG are discussed and verified. Finally, some theories that show the relationship between the Q-CNSG and the soft group are proposed.

Neutrosophic set theory, an advanced framework for error reduction, extends fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs). It enhances effectiveness by refining the definition of indeterminacy, a concept for situations where values cannot be precisely determined. In this paper, we propose dividing indeterminacy into two components based on membership: relative truth (RT), which leans toward truth, and relative falsehood (RF), which leans toward falsehood. This approach improves accuracy by considering, relative truth and relative falsehood, rather than a single indeterminate value. The modified neutrosophic set, called the quadri-portioned neutrosophic soft set (QPNSS), includes four membership attributes: absolute truth, relative truth, relative falsehood, and absolute falsehood, offering greater clarity in uncertain situations. New operations are introduced on QPNSS, such as the quadri-partitioned neutrosophic soft set, subsets, complement, absolute set, set difference, and null set. AND and OR operations are also defined. Additionally, a quadri-partitioned neutrosophic soft topological space (QPNSTS) is defined, with key results presented. We define and explore new concepts such as pre-open (p-open) sets, interior, and closure. The paper also examines QPNS compactness, reducibility to finite sub-covers, and other important properties like the intersection of QPNS p-closed sets and QPNS p-compact spaces. This work contributes to the theoretical understanding of QPNS spaces, particularly in soft topological spaces.

This paper focuses on the set-oriented operations and set-oriented algebraic structures of soft sets. Relatedly, in this paper, firstly some essential properties of $\alpha$-intersection of soft set are investigated, where $\alpha$ is a non-empty subset of the universal set. Later, by using $\alpha$-intersection of soft set, the notion of set-generated soft subring of a ring is introduced. The generators of soft intersections and products of soft subrings are given. Some related properties about generators of soft subrings are investigated and illustrated by several examples.

<abstract><p>Many models of uncertain knowledge have been designed that combine expanded views of fuzziness (expressions of partial memberships) with parameterization (multiple subsethood indexed by a parameter set). The standard orthopair fuzzy soft set is a very general example of this successful blend initiated by fuzzy soft sets. It is a mapping from a set of parameters to the family of all orthopair fuzzy sets (which allow for a very general view of acceptable membership and non-membership evaluations). To expand the scope of application of fuzzy soft set theory, the restriction of orthopair fuzzy sets that membership and non-membership must be calibrated with the same power should be removed. To this purpose we introduce the concept of $ (a, b) $-fuzzy soft set, shortened as $ (a, b) $-FSS. They enable us to address situations that impose evaluations with different importances for membership and non-membership degrees, a problem that cannot be modeled by the existing generalizations of intuitionistic fuzzy soft sets. We establish the fundamental set of arithmetic operations for $ (a, b) $-FSSs and explore their main characteristics. Then we define aggregation operators for $ (a, b) $-FSSs and discuss their main properties and the relationships between them. Finally, with the help of suitably defined scores and accuracies we design a multi-criteria decision-making strategy that operates in this novel framework. We also analyze a decision-making problem to endorse the validity of $ (a, b) $-FSSs for decision-making purposes.</p></abstract>

The behavioural location theory emphasises high importance of the limited rationality and the subjective perception of space in selecting of the location for a business activity. The article discusses key competencies from the scope of behavioural location theory. Ac-cording to the Authors, the behavioural location theory is rather complementary than competitive in relation to the neoclassical or modern approach, as it allows to explain the deviations of the decision-makers from the optimisation behaviour. <b>Business location theory has already been discussed in this journal in various contexts, for example in articles by H. Godlewska-Majkowska, K. Kuciński, A. Rutkowska-Górak, A. Kałowski. However, to our knowledge, the behavioural approach has not yet been presented here and we would like to fill the gap and offer a review of selected authors’ works and concepts from this field and hopefully inspire other scholars to develop this promising research direction. In addition, a quantitative analysis of publications on behavioural location theory will be presented.</b> For the purposes of this article we will define the behavioural location theory as the inclusion of psychological and subjective circumstances of the decision makers into location theory, such as bounded rationality, heuristics usage and subjective spatial perception. The behavioural approach seems underutilised in location theory despite its potential to explain many of business location decisions which are inconsistent with the profit maximisation principle. According to R. Domański [1995]:<i> so far it has not been satisfactorily examined how the perception of space influences spatial behaviour of people. Nobody objects that many decisions, at least in part, depend on how people perceive the space surrounding them, how they differentiate it and what value they place on different elements of this space.</i>According to W. Dziemianowicz [1997]:<i> the assessment of location factors by decision makers most often depends on specific qualities of the business and qualities of the decision maker. </i>Surprisingly, decades have passed since last important contributions in the field of behavioural location theory. Location theory has its roots in XIX century, when J.H. von Thuenen offered the agricultural activity location theory in 1826. The interest in location theory revived more than 50 years later, mainly thanks to the works of W. Launhardt [1882] and A. Marshall [1886]. Important dates are also 1909, when A. Weber developed his industrial location theory and proposed the notion of a location factor and 1933, when the first theory of services location emerged, authored by W. Christaller. Then the development of location theory accelerated, with contributions of such authors as A. Loesch [1939], F. Perroux [1964] or P. Krugman [1991]. It can be argued that thanks to P. Krugman location theory entered the mainstream economics, which neglected spatial issues for a long time. Different location theory traditions put the emphasis on different aspects. For example, classical approach theorists indicate minimising production cost as the goal of the location decision maker while behavioural approach suggests satisfactory choice as a goal. According to H. Godlewska-Majkowska, there are five approaches to location theory: classical, neo-classical, structural, behavioural and contemporary. Their focus points are briefly explained in Table 1. There are three similar but distinct terms related to the business location choice: <br>• location factors – specific qualities of particular places which have direct impact on investment volume during building of the company’s plant (plants) and the net profitability of business activity run in those places [Godlewska-Majkowska, 2001],</br><br> • location virtues – specific qualities of places which contribute to it that identical investments will differ depending on location in terms of investment volume, total production cost, sales revenue and taxes [Godlewska-Majkowska, 2015], ocation circumstances – internal and external phenomena which transform a location virtue into a location factor. Internal phenomena can be for example: industry, size and ownership structure of the business. External phenomena include among others economic, environmental and cultural issues [Godlewska-Majkowska, 2013]. Clearly, location requirements are different for various sectors. Therefore, location factors are divided into general (those applying to all or many sectors) and sector-specific (those applying to one or few sectors). There are also other classifications of location factors. The importance of subjective factors in the location choice is reflexed in the classification by Grabow et al. [1995] into soft and hard location factors, on the basis of H. GodlewskaMajkowska [2015]. Hard factors are more traditional, have direct influence on business activity and are easily measurable, while soft factors have indirect influence on business activity and are difficult to quantify. It is worth to note that authors of this classification consider both kinds of factors as equally important and find even the soft factors as ones, which can be parametrised, measured and compared. Figure 1 presents the classification in a more detailed way. In our view, Grabow et al. [1995] showed an excessive scepticism when it comes to measurability of some factors. For example, the local government attitude towards investor may be measured by places in investment attractiveness rankings, such as ‘Gmina na 5!’ conducted every year by Institute of Enterprise at Collegium of Business Administration at Warsaw School of Economics. Apart from it, the classification should be considered to be validated as more than 20 years had passed since its publication and made more precise, because as H. GodlewskaMajkowska [2015] points out, some factors seem to overlap – social climate is presented as separate factor than local government attitude towards investor, but in fact the former includes the latter. null null Each business has to choose its location and the effect of business location selection is called location decision. Location decision may be the result of a more or less formal procedure. There is a consensus among scholars that business location decision is important for entity’s economic performance. At the same time, it is acknowledged in the literature that subjective factors (such as bounded rationality) play a non-negligible role in location choice. As R. Domański [2004] outlines, location decision makers <i>usually have limited knowledge and incomplete information and in many cases the decision maker does not behave like the</i> homo oeconomicus.<i> Sometimes he has limited or biased information about his decision situation and at the same time he assessed the incomplete information in a subjective way.If the situation is complicated, he has to simplify it by using intuitional rules in decision making. He does not try to achieve the optimal result but rather a satisfactory one. </i>Such statements suggest R. Domański finds bounded rationality model convincing. According to classical, neoclassical and contemporary business location theory the decision maker undertakes the optimal choice, while heterodox approaches such as behavioural location theory claim making an optimal choice is impossible. The classical, neoclassical and contemporary theorists assume decision makers are <i>homo oeconomicus</i>, a person with perfect information about the present and the future, able and willing to make complicated calculations and not prone to psychological biases. Behavioural economics accepts different set of assumptions about the human nature: limited (imperfect) knowledge of the decision maker, limited ability to process the knowledge and searching for satisfactory result rather than optimal. Decision maker who behaves in line with those assumptions is purposefully called <i>homo satisfaciendus</i>. null <i>Homo satisfaciendus</i> is the concept of decision maker used in the bounded rationality model created by H. Simon [1955], which is fundamental for behavioural economics, including behavioural location theory. In the model it is assumed that the decision makers do not aim to maximise utility from choice made (making an optimal decision) but rather search for a good enough (satisfactory) option and once they find such an option they also stop search. In practice, it means that typically a decision maker will accept the first location that meets his minimum criteria, the so called aspiration level and will not even check alternative locations. Simon points out that people may use so called heuristics, which are decision making patterns simplifying their decision problems but he did not elaborate on it. null The gap has been filled by D. Kahneman and A. Tversky [1975] who singled out three famous heuristics: availability, representativeness and anchoring. H. Godlewska-Majkowska [2016] argues that such heuristics are used to assess location virtues of places which a location decision maker had visited within business location decision making process. The bounded rationality model has served as the basis for the A. Pred [1967] behavioural matrix, which linked information availability, investor’s information processing ability and profitability of chosen business location. The general rule is that the more information (or information processing ability) one has, the more profitable location one chooses, <i>caeteris paribus</i>. An adapted version of Pred matrix is presented in Figure 2. Point A represents <i>homo oeconomicus</i>, who has perfect information and perfect ability to use it, so he or she will choose the optimal location. All other decision makers make suboptimal decisions and the extreme is reached in point B, where the decision maker has little information and low ability to process it, so he or she will choose a poor location that may result in a loss.

LetMbe any bounded set inn-dimensional Euclidean space. Then almost alln-dimensional latticesLwith determinant1have the following property: There exists a diagonal transformationDwith determinant1(depending onL) such thatLdoes not cover space withDM. Moreover, ifMhas non-empty interior, the exceptional (null-) set contains at least enumerably many diagonally non-equivalent lattices.

These days, multi-criteria decision-making (MCDM) approaches play a vital role in making decisions considering multiple criteria. Among these approaches, the picture fuzzy soft set model is emerging as a powerful mathematical tool for handling various kinds of uncertainties in complex real-life MCDM situations because it is a combination of two efficient mathematical tools, namely, picture fuzzy sets and soft sets. However, the picture fuzzy soft set model is deficient; that is, it fails to tackle information symmetrically in a bipolar soft environment. To overcome this difficulty, in this paper, a model named picture fuzzy bipolar soft sets (PRFBSSs, for short) is proposed, which is a natural hybridization of two models, namely, picture fuzzy sets and bipolar soft sets. An example discussing the selection of students for a scholarship is added to illustrate the initiated model. Some novel properties of PRFBSSs such as sub-set, super-set, equality, complement, relative null and absolute PRFBSSs, extended intersection and union, and restricted intersection and union are investigated. Moreover, two fundamental operations of PRFBSSs, namely, the AND and OR operations, are studied. Thereafter, some new results (De Morgan’s law, commutativity, associativity, and distributivity) related to these proposed notions are investigated and explained through corresponding numerical examples. An algorithm is developed to deal with uncertain information in the PRFBSS environment. To show the efficacy and applicability of the initiated technique, a descriptive numerical example regarding the selection of the best graphic designer is explored under PRFBSSs. In the end, concerning both qualitative and quantitative perspectives, a detailed comparative analysis of the initiated model with certain existing models is provided.

The concept of N-soft set has been introduced in previous studies where the concept is a development of fuzzy sets and soft sets. In its application, the concept of the N-soft set can only be done if the data from the case to be solved are in the form of non-negative integers only. However, in everyday life, a case is not always obtained in the form of integers. This article provides an approach where data from non-integer cases can be solved. This matter opens the opportunity for studying a more general concept than N-soft set by theoretical.

- Quine's commitment to nominalism has always required set theory to be replaced by an ontologically less dubious tool for the analysis of predication, one that is usually ABSTRACT Rivista di storia della filosofia, n. 1, 2009 mereological in nature, akin to the Calculus of Individuals he and Goodman developed in a seminal paper on nominalism. The problem Quine himself always acknowledged as central in any such replacement arises with the mereological "translation" of numbers. I show here that Quine's proposed translation strategy, even when successful, raises even more serious issues since it requires mereology to supply a substitute for the empty set. After proving the indispensability, given Quine's translation, of such an entity, I demonstrate that an exact mereological replica of the empty set (the "null object") cannot be admitted without engendering contradictions. I conclude by discussing some paraphrases of the usual mereological axioms that prima facie might seem to be compatible with the null object and assessing their implausibility.

<p>In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.</p>

In this paper, we merge the concepts of soft set theory and a bipolar intuitionistic fuzzy set. A bipolar intuitionistic fuzzy soft ideal in a BCK-algebra is defined as a soft set over the set of elements in the BCK-algebra, with each element associated with an intuitionistic fuzzy set. This relationship captures degrees of uncertainty, hesitancy, and non-membership degrees within the context of BCK-algebras. We investigate basic operations on bipolar intuitionistic fuzzy soft ideals such as union, intersection, AND, and OR. The intersection, union, AND, and OR of two bipolar intuitionistic fuzzy soft ideals is a bipolar intuitionistic fuzzy soft ideal. We also demonstrate how to use a bipolar intuitionistic fuzzy soft set to solve a problem involving decision making. Finally, we provide a general approach for handling decision-making problems using a bipolar intuitionistic fuzzy soft set.

We study the existence of a non-spacelike isometry, ζ, in higher-dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of constraints for the metric functions in each case. Within the class of N-dimensional CSI Kundt spacetimes, admitting a non-spacelike isometry, we determine which of these can admit a covariantly constant null vector that also satisfy ζ[a;b] = 0.

This paper focuses on two very important questions: “what is the future of a hybrid mathematical structure of soft set in science and social science?” and “why should we take care to use hybrid structures of soft set?”. At present, these are the most fundamental questions; which encircle a few prominent areas of mathematics of uncertainties viz. fuzzy set theory, rough set theory, vague set theory, hesitant fuzzy set theory, IVFS theory, IT2FS theory, etc. In this paper, we review connections of soft set theory and hybrid structures in a non-technical manner; so that it may be helpful for a non-mathematician to think carefully to apply hybrid structures in his research areas. Moreover, we must express that we do not have any intention to nullify contributions of fuzzy set theory or rough set theory, etc. to mankind; but our main intention is to show that we must be careful to develop any new hybrid structure with soft set. Here, we have a short discussion on needs of artificial psychology and artificial philosophy to enrich artificial intelligence.

The combination of two elements in a group structure is an element, while, in a hypergroup, the combination of two elements is a non-empty set. The use of hypergroups appears mainly in certain subclasses. For instance, polygroups, which are a special subcategory of hypergroups, are used in many branches of mathematics and basic sciences. On the other hand, in a multi-fuzzy set, an element of a universal set may occur more than once with possibly the same or different membership values. A soft set over a universal set is a mapping from parameters to the family of subsets of the universal set. If we substitute the set of all fuzzy subsets of the universal set instead of crisp subsets, then we obtain fuzzy soft sets. Similarly, multi-fuzzy soft sets can be obtained. In this paper, we combine the multi-fuzzy soft set and polygroup structure, from which we obtain a new soft structure called the multi-fuzzy soft polygroup. We analyze the relation between multi-fuzzy soft sets and polygroups. Some algebraic properties of fuzzy soft polygroups and soft polygroups are extended to multi-fuzzy soft polygroups. Some new operations on a multi-fuzzy soft set are defined. In addition to this, we investigate normal multi-fuzzy soft polygroups and present some of their algebraic properties.

AbstractGeneralized relational theories with null values in the sense of Reiter are first-order theories that provide a semantics for relational databases with incomplete information. In this paper we show that any such theory can be turned into an equivalent logic program, so that models of the theory can be generated using computational methods of answer set programming. As a step towards this goal, we develop a general method for calculating stable models under the domain closure assumption but without the unique name assumption.

Hypersoft set theory is an extension of soft set theory and is a new mathematical tool for dealing with fuzzy problems; however, it still suffers from the parametric tools’ inadequacies. In order to boost decision-making accuracy even more, a new mixed mathematical model called the bipolar hypersoft set is created by merging hypersoft sets and bipolarity. It is characterized by two hypersoft sets, one of which provides positive information and the other provides negative information. Moreover, some fundamental properties relative to it such as subset, superset, equal set, complement, difference, relative (absolute) null set and relative (absolute) whole set are defined. Furthermore, some set-theoretic operations such as the extended intersection, the restricted union, intersection, union, AND-operation and OR-operation of two bipolar hypersoft sets with their properties are discussed and supported by examples. Finally, tabular representations for the purposes of storing bipolar hypersoft sets in computer memory are used.

This paper defines the following terms: intuitionistic fuzzy soft (IFS) pretopological space, IFS interior function, IFS pre-open set, IFS pre-closed set, trace of a IFS pretopology, IFS separation axioms, IFS subspace, IFS compactness, IFS connectedness, and some of their properties. In addition, the degree of soft non-vacuity, the soft α-cut, and the IFS preneighbourhood system at a soft point are defined. IFS preneighbourhoods produce IFS pretopologies.

Based on the test section on the expressway in Jiangsu, the thesis makes a model and then conducts numerical simulation on the settlement of soft soil subgrade by using FLAC3D. The author set the filling layer null, and assigns a value to the loaded soil layer according to the construction sequence to realize multi-stage loading. The thesis also analyzes the pore water discharge and foundation deformation when the soil is consolidated. It shows that the simulation result is very close to actual measurement result, and can clearly demonstrate the real situation of settlement.

We are now touching a problem how we add soft computing aspects to logic programming and we have been discussing null attribute values on logic programs. Here, we introduce two functors setu and sets into logic programs for describing indefinite attribute values explicitly. Every logic program with setu or sets has variability and tolerance in itself, namely this program expresses a set of possible definite logic programs. We call this program a variational logic program. In this paper, we show the variational logic programs and a theorem prover for them.

In this paper, we define fuzzy soft ordered Banach algebra with fuzzy soft algebra cone, and introduce the character on fuzzy soft ordered Banach algebra in both cases real and complex. Also, we deduce some of its basic properties and we define a new concept which is a maximal fuzzy soft algebra cone and showing that the set of all fuzzy soft character is isomorphism to the set of all maximal fuzzy soft algebra cone. we prove that the set of all real characters is convex and extreme point, we applied Gelfand- Mazur theorem on fuzzy soft Banach algebra, we showed that character (the set of all complex continuous character) is fuzzy soft ordered Banach algebra. with inverse –closed algebra cone ̆ and a non-zero element in ̆ has inverse we have is an isomorphism to Banach space ( ̆)

It is proved that a non-archimedean FK-space E includes the space c0 of null-sequences if and only if E includes the space l1` of absolutely summable sequences, and for both inclusions the boundedness in E of the set {ek | k ∈ N} of the unit vectors is necessary and sufficient.It is shown that in the non-archimedean context the concepts of wedge space and semiconservative space are equivalent.

The original rough set model, developed by Pawlak depends on a single equivalence relation. Qian et al, extended this model and defined multigranulation rough sets by using finite number of equivalence relations. This model provide new direction to the research. Recently, Shabir et al. proposed a rough set model which depends on a soft relation from an universe V to an universe W . In this paper we are present multigranulation roughness based on soft relations. Firstly we approximate a non-empty subset with respect to aftersets and foresets of finite number of soft binary relations. In this way we get two sets of soft sets called the lower approximation and upper approximation with respect to aftersets and with respect to foresets. Then we investigate some properties of lower and upper approximations of the new multigranulation rough set model. It can be found that the Pawlak rough set model, Qian et al. multigranulation rough set model, Shabir et al. rough set model are special cases of this new multigranulation rough set model. Finally, we added two examples to illustrate this multigranulation rough set model.

AbstractThis paper proposes a new representation for the Public Help Index θ (briefly, PHI θ). Based on winning coalitions, the PHI θ index was introduced by Bertini et al. in (2008). The goal of this article is to reformulate the PHI θ index using null player free winning coalitions. The set of these coalitions unequivocally defines a simple game. Expressing the PHI θ index by the winning coalitions that do not contain null players allows us in a transparent way to show the parts of the power assigned to null and non-null players in a simple game. Moreover, this new representation may imply a reduction of computational cost (in the sense of space complexity) in algorithms to compute the PHI θ index if at least one of the players is a null player. We also discuss some relationships among the Holler index, the PHI θ index, and the gnp index (based on null player free winning coalitions) proposed by Álvarez-Mozos et al. in (2015).

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft σ-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points.

<abstract><p>In the philosophy of rough set theory, the methodologies of rough soft sets and rough fuzzy sets have been being examined to be efficient mathematical tools to deal with unpredictability. The basic of approximations in rough set theory is based on equivalence relations. In the aftermath, such theory is extended by arbitrary binary relations and fuzzy relations for more wide approximation spaces. In recent years, the notion of picture hesitant fuzzy relations by Mathew et al. can be considered as a novel extension of fuzzy relations. Then this paper proposes extended approximations into rough soft sets and rough fuzzy sets from the viewpoint of its. We give corresponding examples to illustrate the correctness of such approximations. The relationships between the set-valued picture hesitant fuzzy relations with the upper (resp., lower) rough approximations of soft sets and fuzzy sets are investigated. Especially, it is shown that every non-rough soft set and non-rough fuzzy set can be induced by set-valued picture hesitant fuzzy reflexive relations and set-valued picture hesitant fuzzy antisymmetric relations. By processing the approximations and advantages in the new existing tools, some terms and products have been applied to semigroups. Then, we provide attractive results of upper (resp., lower) rough approximations of prime idealistic soft semigroups over semigroups and fuzzy prime ideals of semigroups induced by set-valued picture hesitant fuzzy relations on semigroups.</p></abstract>

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subsetE⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactlyEbut still admit approximation in energy by smooth functions.

This article investigates the occurrence and distribution of referential null subjects in Middle English. Whereas Modern English is the textbook example of a non-null-subject language, the case has recently been made that Old English permits null subjects to a limited extent, which raises the question of what happens in the middle period. In this article we investigate Middle English using data drawn from thePenn–Helsinki Parsed Corpus of Middle English Proseand the newParsed Corpus of Middle English Poetry, aiming to shed light on the linguistic and extralinguistic factors conditioning the alternation between null and overt subjects. Generalized mixed-effects logistic regression and random forests are used to assess the importance of the variables included. We show that the set of factors at play is similar to that found for Old English, and we document a near-complete disappearance of the null subject option by the end of the Middle English period.

In this paper non-parametric tests for the multiple resrcssion set up which take into account the known permutation symmetry of the variates under the null hypothesis, are suggested. It bas been shown that under permutation symmetry the proposed procedure is more efficient than the standard nonparametric procedure for the multiple regression problem (see e.g. Puri and Sen (1985) ). A special situation where it is further known in advance that the regressions are identical is also considered briefly.

Soft topology studies a structure on the collection of all soft sets on a given set of alternatives (the relevant attributes being fixed). It is directly inspired by the axioms of a topological space. This paper contributes to the theoretical bases of soft topology in various ways. We extend a general construction of soft topologies from topologies on the set of alternatives in two different directions. An extensive discussion with criteria about what a soft counterpart of “topological separability” should satisfy is also given. The interactions of the properties that arise with separability, and of second-countability and its soft counterpart, are studied under the general mechanisms that generate soft topological spaces. The first non-trivial examples of soft second-countable soft topological spaces are produced as a consequence.

In our work, we have defined different types of double-framed fuzzy soft matrix. Operations of addition, multiplication and complement of double-framed fuzzy soft matrix are defined with examples. Some related propositions with proof is also worked out. Finally medical diagnosis problems have been solved using double framed fuzzy soft matrix.

A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Given 0 ≤ p ≤ q ≤ 1, a dynamical system is [Formula: see text] chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. It shows that, for any 0 ≤ p < q ≤ 1 or p = q = 0 or p = q = 1, a dynamical system which is null and [Formula: see text] chaotic can be realized.

We analyze the null controllability properties from the exterior of two parabolic-elliptic coupled systems governed by the fractional Laplacian \((-d_x^2)^s\), \(s\in(0,1)\), in one space dimension. In each system, the control is located on a non-empty open set of \(\mathbb{R}\setminus(0,1)\). Using the spectral theory of the fractional Laplacian and a unique continuation principle for the dual equation, we show that the problem is null controllable if and only if 1/2<s<1. For more information see https://ejde.math.txstate.edu/Volumes/2020/26/abstr.html