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The aim of this paper is to introduce the concept of (∈,∈∨q)-fuzzy ideals of Sheffer stroke Hilbert algebra with respect to t-norm and derive some interesting result. [ABSTRACT FROM AUTHOR]

In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-conorms are of particular significance, as they are applied to model intersection and union, respectively. Furthermore, the existence of a range of these operators allows for the selection of the t-norm or t-conorm that offers the optimal performance, in accordance with the specific context of the system. In this paper, we obtain new t-norms and t-conorms on some important subfamilies of the set of functions from [ 0 , 1 ] to [ 0 , 1 ] . The structure of these families provides a more solid algebraic foundation for the applications. In particular, we define these new operators on the subsets of the functions that are convex, normal, and normal and convex, as well as the functions taking only the values 0 or 1 and the subset of functions whose support is a finite union of closed intervals. These t-norms and t-conorms are generalized to the type-2 fuzzy set framework. [ABSTRACT FROM AUTHOR]

Copyright of JMD: Journal für Mathematik-Didaktik is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper mainly focuses on new methods of constructing triangular norms as well as triangular conorms on a bounded lattice. At first, we propose the definition of a partition element and show that the definition is well-defined. Then, we construct a t-norm (t-conorm) on a bounded lattice with the help of an interior (a closure) operator and a given t-subnorm (t-subconorm). For the sake of clarity, we give some interpretive examples of the new methods and show that the conditions restricting the elements are indispensable in the construction process. Moreover, we discuss the relationships among the construction methods proposed in this paper and the methods that already exist. Finally, we generalize the new construction methods to modified ordinal sum methods by iteration. [ABSTRACT FROM AUTHOR]

In multi-attribute decision making (MADM), complex situations often arise where decision attributes are interval-valued hesitant fuzzy numbers (IVHFNs) and the attributes are interrelated. Traditional decision-making methods may be ineffective in handling such cases, highlighting the practical importance of seeking more effective approaches. Therefore, finding a more effective decision-making approach has important practical significance. By combining the theories of Archimedean S-norms and T-norms, we innovatively propose a multi-attribute decision-making method based on the generalized interval-valued hesitant fuzzy weighted Heronian mean (GIVHFWHM) operator to address the aforementioned issues. Initially, based on the operational laws of IVHFNs and the Heronian mean (HM) operator, we introduce the generalized interval-valued hesitant fuzzy Heronian mean (GIVHFHM) operator and the GIVHFWHM operator. We then examine properties of the GIVHFHM operator, including permutation invariance, idempotency, monotonicity, boundedness, and parameter symmetry. A multi-attribute decision-making model is constructed based on the GIVHFWHM operator. Finally, we validate the proposed model through numerical experiments in MADM. The results demonstrate that the new decision-making method, based on the GIVHFWHM operator, is feasible and effective in handling multi-attribute decision problems involving IVHFNs with interdependent attributes. This approach provides a novel perspective and method for solving MADM problems under interval-valued hesitant fuzzy conditions with interdependent attributes. It enriches the theoretical framework of multi-attribute hesitant decision models and expands the application of the Heronian mean operator within interval-valued hesitant fuzzy environments. This methodology assists decision makers in making more accurate decisions within complex decision-making contexts, enhancing both the scientific rigor and reliability of decision-making processes. [ABSTRACT FROM AUTHOR]

The Choquet integral is a fuzzy measure that serves as an effective aggregation operator for combining a limited number of components into a single set. In 1978, Hamacher introduced the Hamacher t-norm and t-conorm, an expanded version of algebraic t-norms. In this article, we present the aggregation operators for the Choquet integral that utilize the Hamacher t-norms to handle the theory of complex intuitionistic fuzzy values. These operators include the complex intuitionistic fuzzy Hamacher Choquet integral averaging and geometric operators. Additionally, an analysis is conducted on the attributes and special situations of the suggested methodologies. In addition, a novel approach is presented, utilizing newly developed operators for solving multi-attribute decision-making issues with complex intuitionistic fuzzy values. The operational stages of this strategy are thoroughly presented. Finally, we conducted a comprehensive comparison between the proposed methodology and existing approaches, using illustrative examples to validate the effectiveness and demonstrate the advantages of the proposed methods. [ABSTRACT FROM AUTHOR]

The linguistic q-rung orthopair fuzzy ( L q R O F ) set serves as a useful way of presenting uncertain information by offering more space for decision experts. In the present research, we first link the concepts of Hamacher t-norm and t-conorm with the frame of L q R O F numbers to develop and analyze the innovative L q R O F Hamacher operations. Then, following the proposed Hamacher's norm operations, a series of aggregation operators including L q R O F weighted averaging, L q R O F ordered weighted averaging, L q R O F hybrid averaging, L q R O F weighted geometric, L q R O F ordered weighted geometric, L q R O F hybrid geometric operators are investigated. Some interesting aspects of these AOs are also presented. We further develop evaluation based on distance from average solution (EDAS) approach in light of the newly outlined concepts to cope with L q R O F decision-making problems where the weight information of criteria is fully unknown, ultimately, the practicality of the framed approach is demonstrated through an empirical case, and a detailed analysis is carried out to showcase the methodology dominance. [ABSTRACT FROM AUTHOR]

In this article, we investigate the algebraic structures of the partial orders induced by uninorms on a bounded lattice. For a class of uninorms with the underlying drastic product or drastic sum, we first present some conditions making a bounded lattice also a lattice with respect to the order induced by such uninorms. And then we completely characterize the distributivity of the lattices obtained. [ABSTRACT FROM AUTHOR]

A single-valued Neutrosophic hesitant fuzzy set (SVNHFS) is a combination of a singlevalued neutrosophic set (SVNS) and hesitant fuzzy set (HFS) that has been developed to address insufficient, unreliable, and vague environments in which each element has several possible options determined by the truthiness (𝓉ᴎ), indeterminacy (𝒾ᴎ) and falsity (ẝᴎ) value. By considering this, in this paper, we have proposed the Aczel-Alsina aggregation operator (AAAO) for SVNHFS, which is more flexible t-norm (Ŧ) and t-conorm (𝛻) than the other Ŧ, and 𝛻 due to the flexible nature of parameters to solve Multi-Attribute decision making (MADM) problems. Further, the score function (ȿ), accuracy function (ɑ), and certainty function (ϲ) of SVNHFS have been defined. In this paper, we proposed the single-valued neutrosophic hesitant fuzzy Aczel-Alsina Weighted Averaging Operator (SVNHFAAWA), single-valued neutrosophic hesitant fuzzy Aczel-Alsina Weighted ordered Averaging Operator (SVNHFAAWOA), and single-valued neutrosophic hesitant fuzzy Aczel-Alsina hybrid averaging operator (SVNHFAAHA). To testify to the reliability and stability of the newly created aggregation operator (AO), an application of MADM has been discussed. [ABSTRACT FROM AUTHOR]

Fuzzy relational music perception concerns the representation of congruent connections between musical features as fuzzy relations used to individuate and assemble concepts and conceptual hierarchies. This article presents two universal fuzzy domains of discourse, harmony H and grouping G, which partition sets using triangular norms (t-norms) based on generalised harmonic root support and generalised time regularity, respectively. Fuzzy relations between the sets of the domains are formed in the innate fuzzy neural architecture of a dedicated music faculty. Fuzzy relations are shown to be necessary representations for interconnection between the domains to individuate and assemble concepts. Concepts are individuated and assembled by virtue of fuzzy set resemblance relations between domains, or fuzzy logical implication relations in one or both domains through time. Fuzzy resemblance relations comprise the properties of weak reflexivity, weak symmetry and antitransitivity in a H X G Cartesian product space. Fuzzy implication relations involve fuzzy overlap (or continuation) of elements, calculated using a t-norm operator (min operator), in one or both domains of the product space. Supplementary theory is incorporated to explain polyphonic structure, involving pluralistic superimposition of independent fuzzy relational hierarchies. Broadly, fuzzy relational music perception is a rationalistic model that builds on generative theories and associative--statistical and connectionist approaches by providing a compact and coherent process for determining interaction across musical parameters. [ABSTRACT FROM AUTHOR]

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on continuous fuzzy relational equations programming problems. The fuzzy relational programming problem is a minimization (maximization) problem with a linear objective function subject to fuzzy relational equalities or inequalities defined by certain algebraic operations. In the literature, the commonly seen frameworks for such optimization models are to assume that the operation takes minimum t-norm, strict continuous t-norms (e.g., product t-norm), nilpotent continuous t-norms (e.g., Lukasiewicz t-norm) or Archimedean continuous t-norms. Based on new concepts called partial solution sets, the current paper considers this problem in the most general case where the fuzzy relational equality constraints are defined by an arbitrary continuous t-norm and capture some special characteristics of its feasible domain and the optimal solutions. It is shown that the current generalized results are automatically reduced to (apparently) different ones that hold for special operators when continuous t-norm is replaced by strict, nilpotent or Archimedean continuous t-norm. Also, the relationship between the results derived here and those of previous publications regarding this subject is also discussed. Finally, the proposed algorithm is outlined and illustrated by a numerical example where the continuous fuzzy relational equations is defined by Mayor-Torrens operator that is not an Archimedean t-norm (and then, neither strict nor nilpotent). [ABSTRACT FROM AUTHOR]

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f (x) = lim n → ∞ x T (n) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = TM. [ABSTRACT FROM AUTHOR]

The concepts of cubic spherical neutrosophic sets (CSNSs), introduced and investigated by Gomathi et al. [5], offer a geometric representation of collection of neutrosophic sets (NSs), enhancing their ability to capture uncertainty. The formulation characterizes information using points on a sphere with a defined center and radius, providing a more precise depiction of fuzziness inherent in uncertain data. The cubic spherical neutrosophic Archimedean triangular norms(ATN) and conorms (ATCN), expanding the model's capabilities to handle uncertainty. These algebraic operators enable the aggregation and combination of uncertain information, offering a more comprehensive approach to decision-making. The research further presents a method for solving multiple-criteria decision-making problems within the cubic spherical neutrosophic context, leveraging the newly integrated norms and conorms. The algorithm utilizes the cosine similarity measure of cubic spherical neutrosophic sets, exemplified through an application involving the selection of the most effective electric truck. This extended framework provides decision-makers with enhanced tools to navigate complex decision landscapes amidst uncertainty, facilitating more informed and robust choices across diverse domains. [ABSTRACT FROM AUTHOR]

In this article, we deal with the algebraic structures of the partial orders induced by non-divisible t-norms. We first give a condition for the partial orders induced by non-divisible t-norms being a meet semi-lattice. Then, we show the condition for the partial orders induced by right-continuous t-norms being a lattice, which is very useful for constructing new lattice structures. Finally, an illustrative example is presented. Our results go a further step towards answering the first open problem raised by Karaçal and Kesicioğlu (Kybernetika 47:300-314, 2011). [ABSTRACT FROM AUTHOR]

In this work, we introduce a different interpretation of the notion of a partial fuzzy metric, which we refer to as a partial fuzzy co-metric. We define a partial fuzzy co-metric from a t-conorm and compare it with partial fuzzy metric, in contrast to the conventional approach to the theory of partial fuzzy metric spaces, which is based on the use of a t-norm. Here, we limit the scope of our analysis to Sedghi's definition of partial fuzzy metrics. Additionally, we proposed and compared the ideas of strong partial fuzzy co-metric spaces and strong partial fuzzy metric spaces. We also presented a few examples of these novel ideas. [ABSTRACT FROM AUTHOR]

The primary aim of this article is to put forward new classes of uni-nullnorms on certain classes of bounded lattices via closure (interior) operators. Due to the new classes of uninorms combining both a t-norm T and a t-conorm S by various kinds of closure operators or interior operators, the relationships and properties among the same class of uninorms on L, we obtain new classes of uni-nullnorms on L via closure (interior) operators. The constructions of uninullnorms on some certain classes of bounded lattices can provide another different perspective of t-norms and the dual of t-norms, uninorms and some other associative aggregation operations on bounded lattices. That is, the constructions seem to be the ordinal like sum constructions, but not limited to the ordinal like sum constructions. [ABSTRACT FROM AUTHOR]

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f (x) = lim n → ∞ x T (n) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = TM. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Migrativity is a recently studied property of binary operations defined on the unit interval, introduced by Durante and Sarkoci for studying convex combinations of a continuous t-norm and a drastic product t-norm T D . Inspired by the thought of it, in this paper, we introduce migrativity of extended general binary operations on fuzzy truth values by Zadeh extension principle, where a slight modification is considered. Based on the migrativity equation for fuzzy truth values, we discuss and present some of its characterizations specific to the binary operation that is migrative over a class of particular fuzzy truth values related to characteristic functions of elements in [0, 1] and then extend it to the rather general cases, which leads to the connections and equivalence between two concepts of migrativity under specific conditions. The results of this paper can be applied immediately to the extensions of some familiar aggregation functions like t-norms and overlap functions, revealing the preliminary traits of migrativity on type-2 set. [ABSTRACT FROM AUTHOR]

Variable precision fuzzy rough set (VPFRS) is widely utilized for handling various forms of uncertain information due to its fault-tolerant capability. However, a significant number of these rough sets fail to satisfy the inclusion property (lower approximation included in the upper approximation), posing potential risks in applications. Moreover, a common method of constructing the VPFRS is through triangular norms and triangular conorms. But in certain practical applications, the associative law of triangular norms and triangular conorms may not be essential. Overlap functions and grouping functions can effectively avoid this issue. Therefore, to address the limitations of existing models, we introduce the concept of VPFRS based on overlap and grouping functions, and apply it to a real multi-attribute decision-making problem. Firstly, we propose a novel VPFRS leveraging overlap and grouping functions, and demonstrate that it satisfies the generalized inclusion property. This solves the deficiency in VPFRSs not meeting the inclusion property to some extent. Additionally, with the help of the generalized inclusion property, we introduce a new objective method for computing attribute weights. Subsequently, by integrating the merits of the proposed VPFRS model and the PROMETHEE method, we develop a multi-attribute decision-making method with double weight. Finally, the validity of our decision-making method and weight calculation approach is substantiated through comparison and experimental analysis. [ABSTRACT FROM AUTHOR]

Introduction: The triangular recess (TR), also called triangular fossa or vulva cerebri, represents the anterior extension of the diencephalic ventricle, located between the anterior columns of the fornix and the anterior white commissure. Over time, this structure of the third cerebral ventricle generated many disputes. While some anatomists support its presence, others have opposite opinions, considering that it only becomes visible under certain conditions. The aim of the study is to demonstrate the tangible structure of the triangular recess. Secondly, the quantitative analysis allowed us to establish an anatomical morphometric standard, as well as the deviations from the standard. Materials and methods: Our study is both a quantitative and a qualitative evaluation of the triangular fossa. We dissected 100 non-neurological adult brains, which were fixed in 10% formaldehyde solution for 10 weeks. The samples are part of the collection of the Institute of Anatomy, "Grigore T. Popa" University of Medicine and Pharmacy, Iasi. We highlighted the triangular fossa by performing dissections in two stages at the level of the roof of the III ventricle. Results: The qualitative analysis is a re-evaluation of the classical data concerning the anatomy of the fossa triangularis. We proposed an original 3D model of the triangular recess in which we described a superficial part called vestibule and a deep part called pars profunda. We measured the sides of the communication between the two proposed segments, as well as the communication with the III ventricle. By applying the Heron's formula, we calculated the area of the two communications. Statistical evaluations have shown that these communications are higher than they are wide. In addition, there is a statistical difference between the surfaces of the two communications: 34.07 mm² ± 7.01 vs. 271.43 mm² ± 46.36 (p = 0.001). Conclusion: The outcome of our study is both qualitative and quantitative. Firstly, we demonstrated the existence of the triangular fossa and we conceived a spatial division of this structure. Secondly, the measurements carried out establish an anatomo-morphometric norm of the triangular recess, which is useful in assessing the degree of hydrocephalus during the third endoscopic ventriculoscopy. [ABSTRACT FROM AUTHOR]

Semi-overlap functions as a generalization of left-continuous t-norms also have residua. In this paper, we develop a new residuated logic, SOL-logic, based on semi-overlap functions and their residua. The corresponding algebraic structures, SOL-algebras, are defined, and the completeness of SOL with respect to SOL-algebras is proved. [ABSTRACT FROM AUTHOR]

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/ hp –version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p , we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime. [ABSTRACT FROM AUTHOR]

We explore a new fuzzy metric constructed from already defined fuzzy metrics over the same set using pseudo-linear combination. Operations used in pseudo-linear combination are triangular norm and conorm. The fuzzy space thus obtained is proved to be complete. Additional features related to this space are also presented. A fuzzy metric obtained in this way can be used to construct an image denoising procedure, from the fuzzy metrics used for the spatial distance and the color similarity measure between the pixels in the image. The goal is to enhance the sharpness and quality of the image, expressed and measured by the image quality index. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy (IF) sets are valuable tools for describing uncertain information in Multi-Criteria Group Decision Making (MCGDM), where the elements have degrees of membership and non-membership. IF aggregation operator is a popular data processing method that can be used for data dimensionality reduction, feature extraction, data compression, and so on. Some existing MCGDM techniques based on IF aggregation operators have been criticized for reasons that include disregarding the comprehensive correlations of the criteria and ignoring the monotonicity of the decision information. This paper aims to construct some IF aggregation operators based on Yager's triangular norms and Heronian mean to shed light on decision-making issues. At first, some novel IF operations such as Yager sum, Yager product, and Yager scalar multiplication on IFSs are presented. Based on these new operations, the generalized IF Yager Heronian average (GIFYHA) operator and the generalized IF Yager weighted Heronian average (GIFYWHA) operator are proposed and their corresponding properties are also proved in detail. Then, an improved MCGDM algorithm is constructed that relies on suggested operators. Its effectiveness and applicability are verified by applying it to select the best location for a company. In addition, the sensitivity of the parameters in the proposed operator to decision findings is also discussed. Finally, the comparative analysis of the proposed operator with the existing operators shows that the proposed operator is suitable for aggregating IF information with correlations both on "non-empty lattice" and total orders on IF values. [ABSTRACT FROM AUTHOR]

This article aims to contribute to the theory of ordinal sum implications by introducing two new classes: the minor ordinal sum implications and the major ordinal sum implications. The study presents diverse construction methods employed in generating examples of these two new classes. A method of constructing ordinal sums of implications that may be neither major nor minor is also presented. Furthermore, the article extends its contribution by providing examples of major and minor ordinal sum implications within some of the main classes of fuzzy implications, including (S,N)-implications, (T,N)-implications, QL-implications and D-implications. A study of the properties satisfied by the new ordinal sum implications is also presented. [ABSTRACT FROM AUTHOR]

Online image theft is a severe concern faced by many photographers. It occurs when someone uses the photo they took accidentally or deliberately without getting the permission of the owner of the photo. One of the solutions to prevent online image theft is to use biometric watermarking on the photo to be protected. The proposed work integrates multimodal biometric watermarking, Lifting Wavelet Transform (LWT), and score-level fusion using Triangular Norms to achieve better accuracy. Using block-based representatives from LWT coefficients for embedding preserves the visual quality of the watermarked image. By considering the iris and fingerprint images from the multimodal biometric database, SDUMLA-HMT, the proposed approach tests the watermarked images against many attacks and is proven to attack resilience. Moreover, the proposed system tests watermark imperceptibility and are proven not to affect the watermarked image's visual quality. The proposed system has also achieved a better accuracy rate regarding low False Acceptance Rate and False Rejection Rate. [ABSTRACT FROM AUTHOR]

Uninorms are a special type of associative aggregation functions, which have received widespread attention in the theoretical and practical fields since their introduction. Durante and Sarkoci introduced the migrativity property in 2008. Afterwards, this property was widely applied in numerous fields like image processing and decision analysis, which has sparked a series of studies. There have been a large number of research results on the migrativity involving uninorms, but the work has mainly focused on the uninorms internal on the boundary. In this paper, we will concentrate on the uninorms not internal on the boundary. First, we discuss the characterization of the α-migrativity of conjunctive uninorms over continuous t-norms according to the value of α. Then, the consequences of the α-migrativity of disjunctive uninorms over continuous t-conorms can be obtained dually. [ABSTRACT FROM AUTHOR]

Aggregation operators have the important application in any fields where the fusion of information is processed. The dominance relation between two aggregation operators is linked to the fusion of fuzzy relations, indistinguishability operators and so on. In this paper, we deal with the weak dominance relation between two aggregation operators which is closely related with the dominance relation. Weak domination of isomorphic aggregation operators and ordinal sum of conjunctors is presented. More attention is paid to the weak dominance relation between ordered weighted averaging operators and Lukasiewicz t-norm. Furthermore, the relationships between weak dominance and some functional inequalities of aggregation operators are discussed. [ABSTRACT FROM AUTHOR]

The circular intuitionistic fuzzy set (CIFS) extends the concept of IFS, representing each set element with a circular area on the IFS interpretation triangle (IFIT). Each element in CIFS is characterized not only by membership and non-membership degrees but also by a radius, indicating the imprecise areas of these degrees. While some basic operations have been defined for CIFS, not all have been thoroughly explored and generalized. The radius domain has been extended from [0, 1] to [0, √2]. However, the operations on the radius domain are limited to min and max. We aimed to address these limitations and further explore the theory of CIFS, focusing on operations for membership and nonmembership degrees as well as radius domains. First, we proposed new radius operations on CIFS with a domain [0, ψ], where ψ ∈ [1, √2], called a radius algebraic product (RAP) and radius algebraic sum (RAS). Second, we developed basic operators for generalized union and intersection operations on CIFS based on triangular norms and conorms, investigating their algebraic properties. Finally, we explored negation and modal operators based on proposed radius conditions and examined their characteristics. This research contributes to a more explicit understanding of the properties and capabilities of CIFS, providing valuable insights into its potential applications, particularly in decision-making theory. [ABSTRACT FROM AUTHOR]

Supplier selection is a very crucial process within a business or commercial enterprise because it depends upon different components like reliability, customer need, services, cost and reputation. A suitable supplier is familiar with developing a relationship between customer needs and business. To serve this purpose, the multiple attribute group decision-making (MAGDM) technique is a well-known and efficient aggregation model used to evaluate flexible optimal options by considering some appropriate criteria or attributes. Experts face some sophisticated challenges during the decision-making process due to uncertain and ambiguous information about human opinions. To address such conditions, we explore the notion of spherical fuzzy sets (SFS) and their reliable operations. Some flexible operational laws of Dombi t-norms are also developed in light of spherical fuzzy (SF) information. Combining the theory of Hamy mean (HM) models and Dombi aggregation tools, some robust strategies are also studied in this research work. The main objectives of this article are to propose some dominant strategies in the presence of SF information including spherical fuzzy Dombi Hamy mean (SFDHM), spherical fuzzy Dombi weighted Hamy mean (SFDWHM), spherical fuzzy Dombi Dual Hamy mean (SFDDHM) and spherical fuzzy Dombi weighted Dual Hamy mean (SFDWDHM) operators. The MAGDM techniques are utilized to evaluate the flexibility of our derived methodologies under considering SF information. An experimental case study is utilized to evaluate a notable supplier enterprise under consideration of our developed methodologies. Finally, a comprehensive overview of our research work is also presented. [ABSTRACT FROM AUTHOR]

Machine learning and image processing are closely related fields that have undergone major development and application in recent years. Machine learning algorithms are being used to develop sophisticated techniques for analyzing and interpreting images, such as object detection, image classification, and image segmentation. One important aspect of image processing is the ability to compare and measure the similarity between different images by providing a way to quantify the similarity between images using various features such as contrast, luminance, and structure. Generally, the flexibility of similarity measures enables fine-tuning the comparison process to achieve the desired outcomes. This is while the existing similarity measures are not flexible enough to address diverse and comprehensive practical aspects. To this end, we utilize triangular norms (t-norms) to construct an inclusive class of similarity measures in this article. As is well-known, each t-norm possesses distinctive attributes that allow for novel interpretations of image similarities. The proposed class of t-norm-based structural similarity measures offers numerous options for decisionmakers to consider various issues and interpret results more broadly in line with their objectives. For more details, in the Experiments section, the proposed method is applied to grayscale and binarized images and a specific experiment related to meteorology. Eventually, the presented diverse case studies confirm the efficiency and key features of the t-norm-based structural similarity. [ABSTRACT FROM AUTHOR]

Rosenfeld defined a fuzzy subgroup of a given group as a fuzzy subset with two special conditions and Mustafa Demirci proposed the idea of fuzzifying the operations on a group through a fuzzy equality and a fuzzy equivalence relation. This paper mainly focuses on fuzzy subsets and vague sets of monoids with several extended algebraic properties. Firstly, we generalize some algebraic properties of t-norms to fuzzy t-norms, this allows for a broader analysis and classification of fuzzy t-norms, enabling their wider application. Furthermore, we explore specific research on the properties of vague t-norms. Finally, selected conclusions about fuzzy t-norms are extended to bounded lattices. [ABSTRACT FROM AUTHOR]

There are some drawbacks to arithmetic and logic operations of general discrete fuzzy numbers, which limit their application. For example, the result of the addition operation of general discrete fuzzy numbers defined by the Zadeh's extension principle may not satisfy the condition of becoming a discrete fuzzy number. In order to solve these problems, special discrete fuzzy numbers on countable sets are investigated in this paper. Since the representation theorem of fuzzy numbers is the basic tool of fuzzy analysis, two kinds of representation theorems of special discrete fuzzy numbers on countable sets are studied first. Then, the metrics of special discrete fuzzy numbers on countable sets are defined, and the relationship between these metrics and the uniform Hausdorff metric (i.e., supremum metric) of general fuzzy numbers is discussed. In addition, the triangular norm and triangular conorm operations (t-norm and t-conorm for short) of special discrete fuzzy numbers on countable sets are presented, and the properties of these two operators are proven. We also prove that these two operators satisfy the basic conditions for closure of operation and present some examples. Finally, the applications of special discrete fuzzy numbers on countable sets in image fusion and aggregation of subjective evaluation are proposed. [ABSTRACT FROM AUTHOR]

In this paper, on a bounded lattice L, we give a new approach to construct uninorms via a given uninorm U* on the subinterval [0, a] (or [b, 1]) of L under additional constraint conditions on L and U*. This approach makes our methods generalize some known construction methods for uninorms in the literature. Meanwhile, some illustrative examples for the construction of uninorms on bounded lattices are provided. [ABSTRACT FROM AUTHOR]

Inspired by the thought of distributivity between semi-t-operator and S-uninorm, this paper primarily explores the distributivity between extended semi-t-operator and extended S-uninorm on fuzzy truth value. First, Zadeh-extended semi-t-operator and S-uninorm are proposed on fuzzy truth value and some results of extended semi-t-operator are studied under special fuzzy truth values. Then, it concentrates on the sufficient condition about left and right distributivity of extended semi-t-operator over extended S-uninorm under the condition that semi-t-operator is left and right distributive over S-uninorm, respectively. Finally, when parameters satisfy different cases, sufficient conditions for the distributivity between extended semi-t-operator and extended S-uninorm are given under the condition that semi-t-operator satisfies distributivity or conditional distributivity over S-uninorm. [ABSTRACT FROM AUTHOR]

This study employs the notions of t-norms and t-co-norms to define a group of T -neutrosophic sub-groups and normal T -neutrosophic subgroups. Furthermore, the different properties of these sub-groups have been investigated. After that, the t-norm and the t-co-norm were applied to the finite direct product of the group. [ABSTRACT FROM AUTHOR]

As a generalization of intuitionistic fuzzy sets, single-valued neutrosophic sets have certain advantages in solving indeterminate and inconsistent information. In this paper, we study the fuzzy inference full implication method based on single-valued neutrosophic t-representable t-norm. Firstly, single-valued neutrosophic fuzzy inference triple I principles for fuzzy modus ponens and fuzzy modus tollens are given. Then, single-valued neutrosophic R-type triple I solutions for FMP and FMT are given. Finally, the robustness of the full implication triple I method based on the leftcontinuous single-valued neutrosophic t-representable t-norm is investigated. As a special case of the main results, the sensitivity of full implication triple I solutions based on three special single-valued neutrosophic t-representable t-norms are given. [ABSTRACT FROM AUTHOR]

Trapezoidal Atanassov's intuitionistic fuzzy numbers (TrAIFNs) is one of the useful tools to manage the fuzziness and vagueness in expressing decision data and solving decision making problems. In this paper, based on the operation laws defined by strict t-norms and t-conorms, four kinds of power geometric operators, i.e., triangular (co)norms-based (T-based) power geometric operator of TrAIFNs, T-based weighted power geometric operator of TrAIFNs, T-based power ordered weighted geometric operator of TrAIFNs, and T-based power hybrid geometric operator of TrAIFNs, are developed. To minimize loss of information in process, a new ranking method of TrAIFNs are presented based on the newly proposed possibility differences of TrAIFNs; Moreover, utilizing strict t-conorms, a new similarity measurement of TrAIFNs is innovated. Thereby, in combination with all the referred elements, two approaches to multiple attributes group decision making using TrAIFNs are developed. In the end, the feasibility of those methods and the superiority over the existing methods are demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

Due to the intense competition in the market today, choosing of an appropriate healthcare device vendor in long-term organ transplant networks has emerged as a key issue in raising life expectancy. A complicated multi-attribute group decision-making (MAGDM) process problem with several viable alternatives and sustainable criteria may be used for evaluating sustainable healthcare equipment vendors. The q-rung orthopair fuzzy set (QROFS) is more effective at expressing ambiguous and fuzzy information since it is a generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFS). The fact that QROFSs provide a wider range of acceptable membership grades and give decision-makers more leeway to express their real thoughts is their most valued feature. The Heronian mean (HM) operator and power aggregation (PA) operator are instances of classic aggregation operators. They are preferable because they can replicate the correlations between attributes and remove the negative effects of awkward information. The Aczel-Alsina t-norms, which were put out by Aczel and Alsina in 1982, constitute a very successful and widely used method for creating any type of aggregation operators. Additionally, the parameter Φ ∈ 0 , + ∞ makes the Algebraic t-norms as a special case of the Aczel-Alsina t-norms. To take the above advantages, in this article, initially, the Aczel-Alsina (AA) operational laws are combined, with power average and Heronian mean operators to propose the QROFAA power Heronian aggregation (QROFPWHA) operator and QROFAA power geometric Heronian aggregation (QROFAAFPGHA) operator. Moreover, some core characteristics and various core cases with respect to the parameters are investigated and found that some of the existing aggregation defined on AA operational laws are special cases of the suggested aggregation operators. Secondly, the weighted forms of the suggested aggregation operators are initiated. Thirdly, based on these newly aggregation operators two novel MAGDM models with unknown weights of the decision makers and attributes are initiated. Finally, an illustrated example about evaluating sustainable healthcare equipment vendors is provided to assess the effectiveness of the suggested models, and a comparison analysis is provided to support and corroborate the suggested approaches. Additionally, assessment for key parameters in the proposed models is carried out to evaluate the implications on results. [ABSTRACT FROM AUTHOR]

Hesitant Fermatean fuzzy sets (HFFS) can characterize the membership degree (MD) and non-membership degree (NMD) of hesitant fuzzy elements in a broader range, which offers superior fuzzy data processing capabilities for addressing complex uncertainty issues. In this research, first, we present the definition of the hesitant Fermatean fuzzy Bonferroni mean operator (HFFBM). Further, with the basic operations of HFFS in Einstein t-norms, the definition and derivation process of the hesitant Fermatean fuzzy Einstein Bonferroni mean operator (HFFEBM) are given. In addition, considering how weights affect decision-making outcomes, the hesitant Fermatean fuzzy weighted Bonferroni mean (HFFWBM) operator and the hesitant Fermatean fuzzy Einstein weighted Bonferroni mean operator (HFFEWBM) are developed. Then, the properties of the operators are discussed. Based on HFFWBM and HFFEWBM operator, a new multi-attribute decision-making (MADM) approach is provided. Finally, we apply the proposed decision-making approach to the case of a depression diagnostic evaluation for three depressed patients. The three patients' diagnosis results confirmed the proposed method's validity and rationality. Through a series of comparative experiments and analyses, the proposed MADM method is an efficient solution for decision-making issues in the hesitant Fermatean fuzzy environment. [ABSTRACT FROM AUTHOR]

Fuzzy rough set models are useful tools for dealing with fuzzy and real-valued data. They have been used in many real-world applications. In this paper, we investigate the fuzzy rough set model based on triangular norms and fuzzy implications. First, we extend some results in the published literature by removing the condition that is the continuity of triangular norms, and obtain more general conclusion about fuzzy upper approximation operators. Then, for the fuzzy neighborhood and the fuzzy lower approximation operator based on fuzzy implications, we investigate their characterization with each other. Finally, we establish the relationships between fuzzy rough sets and fuzzy topology. In this work, researches on the properties of fuzzy rough sets based on triangular norms which need not be continuous provide generalization results for fuzzy rough set theory from viewpoint of mathematics. [ABSTRACT FROM AUTHOR]

Aggregation operators (AOs) are utilized to overcome the effects of attributes under some specific degree of weight in the decision-making (DM) process. The AOs have a large capacity to deal with uncertain and unpredictable information in multi-attribute decision-making (MADM) problems. The Hamy mean (HM) aggregation tools are well-known aggregation models, which are utilized to define correlation among different input arguments adequately. The intuitionistic fuzzy (IF) sets (IFS) can express unpredictable and vague information. The Aczel Alsina aggregation expressions are extensions of triangular norms. Recently, Aczel Alsina aggregation tools attained a lot of attentions from numerous research scholars. By inspiring the robustness and reliability of Aczel Alsina aggregation tools, we expose some appropriate operations of Aczel Alsina expressions under consideration of IF information. In this manuscript, we developed an intuitionistic fuzzy Aczel Alsina HM (IFAAHM) and an intuitionistic fuzzy Aczel Alsina weighted HM (IFAAWHM) operator. We also expressed the theory of Dual HM (DHM) tools and established a series of new approaches including intuitionistic fuzzy Aczel Alsina Dual HM (IFAADHM) and intuitionistic fuzzy Aczel Alsina weighted Dual HM (IFAAWDHM) operators. Some reliable characteristics and special cases of our derived approaches are also presented. The authors solved an application of a MADM technique under consideration of our derived approaches. To check the reliability and dependency of our derived mythologies, we gave an experimental case study to evaluate a desirable construction material based on some specific criteria of different Alternatives. To see the advantages and compatibility of our derived approaches, by comparing the results of existing approaches with the results of currently discussed AOs. [ABSTRACT FROM AUTHOR]

Real-world decision-making problems often include multi-polar uncertainties dependent on multi-dimensional attributes. The m-polar fuzzy (mF) sets can efficiently handle such multi-faceted complications with T-norm based weighted aggregation techniques. The Aczel–Alsina T-norms offer comparatively flexible and accurate aggregation than the other well-known T-norm families. Consequently, this work introduced novel mF Aczel–Alsina aggregation operators (AOs), including weighted averaging (mFAAWA, mFAAOWA, mFAAHWA) and weighted geometric (mFAAWG, mFAAOWG, mFAAHWG) AOs. The fundamental properties, including boundedness, idempotency, monotonicity, and commutativity are investigated. Based on the proposed AOs, a decision-making algorithm is developed and implemented to solve two detailed multi-polar site selection problems (for desalination plant and for wind-power plant). Finally, a comparison with mF Dombi and mF Yager AOs reveals that different T-norm based AOs may yeild different solutions for the same problem. [ABSTRACT FROM AUTHOR]

The α-cross-migrativity can be regarded as weaker form of the commuting equation. It has been extensively investigated between some aggregation functions including t-norms, overlap functions, uninorms, and semi-t-operators. Recently, Fang [10] has proposed the α-cross-migrativity of t-conorms over fuzzy implications. This paper continues to consider this research topic and mainly focuses on the fuzzy implications generated by additive (resp. multiplicative) generators of continuous Archimedean t-norms and t-conorms. Full characterizations for the α-cross-migrativity of continuous t-conorms over (f, g)-, k-, h- and (θ, t)-generated implications are obtained. Moreover, some supporting examples for solutions are given. [ABSTRACT FROM AUTHOR]