Using the *-fuzzy measure introduced by Ghafari, Saadati, and Mesiar [Fuzzy Sets and Systems 2022], we develop the concept of martingale within *-fuzzy measure spaces. In particular, we establish *-fuzzy analogues of a Doob decomposition, Doob's maximal inequality, optional stopping theorems for both bounded and unbounded stopping times, and martingale convergence results. The analysis is carried out at the level of *-fuzzy integrals, reflecting the intrinsic nonlinearity of the model. It is supported by the *-fuzzy conditional expectations and the fuzzy Radon-Nikodym type representations. These results provide a coherent dynamic framework for martingales under *-fuzzy measures.

Recently, Zong et al. demonstrated that a t-norm is an additively generated cancellative t-norm if and only if the fully ordered semigroup induced by it can be order-embedded into the multiplicative semigroup of the unit real interval. In the present study, we put forward several criteria for determining whether a t-norm is an additively generated cancellative one.

The Sugeno integral as a typical non-additive integral, is widely used in decision-making, comprehensive evaluation, and classification problems involving uncertainty or fuzziness. The concept of Sugeno integrals originated from Sugeno, and many scholars have effectively promoted its application. To this day, further development of Sugeno integrals remains a significant research direction. This paper aims to further advance the theory of Sugeno integrals by first introducing a new integral, known as the double set-function Sugeno integral (DSSI). This DSSI extends the original Sugeno integral, which was based on a single fuzzy measure, to one that is based on both set-functions and fuzzy measures. The paper also explores the monotonicity of this integral and its Jensen’s inequality, among other properties. Secondly, it discusses the convergence theorems for two types of integral sequences: those involving set-function sequences and those involving function sequences. It derives several convergence theorems, including the monotone convergence theorems, Fatou’s lemmas, and dominated convergence theorems. Finally, the paper examines discrete DSSI, provides its specific representation, and demonstrates through examples that it outperforms classical Sugeno integrals in decision-making problems.

In this study, we extend the additively generated triangular norms from the framework of the unit interval to that of partially ordered sets. We present several conditions under which this formula yields a t-norm, where are partially ordered sets, and are monotone functions and is a t-norm/t-conorm on . The partially ordered semigroups induced by are order-preserving/order-reversing homomorphic to semigroup deformations of the semigroup induced by .

Granger causality models are widely used in time series analysis to manage causal relationships between variables based on temporal precedence. Traditional Granger causality models always assume linear relationships and normality assumptions, which often fail to capture the complexities of real-world data involving nonlinear and non-normal behavior. In this work, we propose a copula-based approach that offers a novel alternative method by capturing nonlinearity and non-normality in multivariate time series modeling. We develop a copula-based joint regression model tailored for Granger causality analysis and carry out some predictions in these models. We assess and validate the performance of our method using simulation studies in different scenarios. Two applications of our proposal in real-world data analysis are also presented. Our findings highlight the superiority of copula-based approaches compared to other Granger causality models in complex time series data.

In many decision-making systems, relationship among entities are best represented using directed graphs. They capture asymmetric connections among the entities where influence flows in one direction rather than being reciprocated. In such systems, decisions are significantly impacted by variations in the directed edges, as they determine how influence propagates through the entities of the considered network. This study focuses on developing fusion functions that incorporate these directional influences represented through directed edges into the aggregation process, ensuring that the final outcome reflects the structural dependencies and varying levels of influence within the system. The first step is to analyze how different entities in a network are connected and how influence flows through these directed links. Then, to model the aggregation framework, we propose a fusion function well adapted to incorporate the directional nature of influence. This function ensures that aggregation respects the asymmetric relationships in the network, effectively capturing the impact of influence propagation on the final decision. We also extend our approach by developing a pre-aggregation version of the directed graph-based fusion functions. The pre-aggregation function is designed to incorporate the non-symmetric relationships among entities, where aggregation increment is done in a particular direction. This generalization is adaptable for various decision-making systems with different types of relationships. In such aggregation systems, certain properties of fusion functions such as idempotency and averaging behavior, etc. are often desirable. These properties are incorporated into the proposed framework to ensure its usability across diverse decision-making contexts.

The notion of spline linearity for aggregation functions is recalled. We then focus on linear spline 2-copulas and various constructions of 2-copulas to investigate the impact of these constructions on spline linearity. Moreover, associative copulas that are linear splines are discussed. Some other methods that yield linear spline copulas are also presented and illustrated.

In this paper, some new properties of the Sugeno integral and the seminormed fuzzy integrals are shown, and by means of these properties, we investigate the entropy of fuzzy sets (fuzzy entropy) in the sense of De Luca and Termini on continuous domain. In the framework of fuzzy measures, we present a set of necessary and sufficient conditions of the fuzzy entropy defined by Sugeno integral. We provide a counterexample to show that the previously presented results regarding the fuzzy entropies defined by fuzzy integrals are incomplete. Several calculation examples of the fuzzy entropy defined by Sugeno integral are also demonstrated. The above results are extended to the case of the seminormed fuzzy integrals.

Risk allocation in the micromobility sector is challenged by multi-stakeholder participation, interdependent criteria, and high uncertainty in expert judgments, which limits the effectiveness of conventional multi-attribute group decision-making (MAGDM) approaches. This study proposes a data-driven technique for evaluating risk-prevention alternatives in the micromobility sector using a q-rung orthopair fuzzy rough number (q-ROFRN). The primary objective of this research is to develop an integrated CRITIC-REGIME model by combining the Criteria Importance Through Intercriteria Correlation (CRITIC) method and the Rangements Et Gestion Integree des MElanges (REGIME) technique to address MAGDM problems. The proposed approach is applied to a real-world micromobility case study involving multiple governance and operational alternatives evaluated across several dimensions and 18 sub-criteria. The results indicate that state-private sector partnerships represent the most effective strategy for managing risks in micromobility systems. The findings further demonstrate that the proposed framework improves decision-making accuracy by effectively integrating multiple influential factors and accounting for uncertainty in expert assessments. Comparative and sensitivity analyses confirm that the integrated q-ROFRN-based CRITIC-REGIME model outperforms existing decision-making approaches in terms of precision, robustness, and adaptability under uncertain conditions.

We introduce the concept of time-dependent fuzzy measures for modeling systems that evolve over time. It is based on two main approaches: memoryless fuzzy measures and B-dynamical fuzzy measures, where B represents a binary aggregation function. Although memoryless measures depend solely on current data, B-dynamical measures incorporate memory by recursively aggregating the previous state with new incoming information. We analyze these dynamical measures from a mathematical perspective with particular emphasis on their convergence and stability. Using the Lipschitz continuity of aggregation functions, we establish conditions under which the system converges to a unique steady state, independent of its initial measure. Several numerical examples are provided to illustrate how these models adapt to new information. These results provide a theoretical basis for modeling adaptive systems in which the relevance of criteria or information sources changes over time while preserving stable long-term behavior. In addition, the α-weighted mean model offers a linear and intuitively interpretable description of the balance between memory and new information.

This paper studies continuous Archimedean t-norms and copulas representable by linear spline functions. Based on the classical additive generator representation, we show that a continuous Archimedean t-norm is a two-dimensional linear spline if and only if its additive generator is a one-dimensional linear spline. The introduced upper corner condition provides a geometric criterion linking the linearity of the pseudo-inverse function with the structure of the t-norm. These results yield a simple and complete characterization of Archimedean copulas and t-norms expressible through linear splines.

In this paper, compatible congruences with respect to an aggregation operation are discussed. First, we establish the homomorphism theorems for aggregation operations. Second, we investigate a-classes of elements of a bounded lattice (with respect to an aggregation operation and a filter/ideal) introduced by Ince, and show that all these classes form a bounded lattice under set inclusion.

In *-fuzzy Banach spaces equipped with a continuous t-norm *, we construct a *-fuzzy measure of noncompactness, motivated by classical measures of noncompactness and *-fuzzy measures. This approach extends classical noncompactness measures to the fuzzy setting while preserving their essential properties. The proposed measure satisfies eight fundamental axioms (FM1)-(FM8).We establish several key theorems, including the verification that the proposed measure fulfills all axioms. As a main application, we prove the existence of solutions for a class of nonlinear functional integral equations of the formx(γ)=H(γ,x(γ))+∫0γJ(γ,σ,x(σ))dσ,γ∈R+,in the space E=BC(R+;V) (the *-fuzzy Banach space of fuzzy bounded continuous functions).Numerical simulations are presented to support the theoretical results, where the Artificial Small Parameter Method (ASPM) is employed to approximate the solution. The numerical examples confirm the theoretical findings and illustrate the effectiveness of the proposed approach.

Due to its reasonable properties, the Kulisch and Miranker binary relation on the family of all closed and bounded real intervals has attracted the attention of many researchers, especially in the field of Computation. However, it is not total, so there are intervals that are not comparable. To face this problem, Bustince et al. introduced the notion of admissible order, which is coherent to the Kulisch and Miranker binary relation. Due to its technical construction, most of the examples of admissible orders are defined by only employing the extremes of such intervals. In this paper we introduce a non-countable family of admissible orders in the set of all closed and bounded subintervals contained in a concrete closed and bounded real interval. The approach is novel in two senses: on the one hand, due to the mathematical objects that are involved (a dense sequence and a family of continuous functions); and, on the other hand, we do not handle the intervals through their extremes, but only by their interior points.

A useful expansion of the intuitionistic fuzzy set (IFS) for dealing with ambiguities in information is the Pythagorean fuzzy set (PFS), which is one of the most frequently used fuzzy sets in data science. Due to these circumstances, the Aczel-Alsina operations are used in this study to formulate several Pythagorean fuzzy (PF) Aczel-Alsina aggregation operators, which include the PF Aczel-Alsina weighted average (PFAAWA) operator, PF Aczel-Alsina order weighted average (PFAAOWA) operator, and PF Aczel-Alsina hybrid average (PFAAHA) operator. The distinguishing characteristics of these potential operators are studied in detail. The primary advantage of using an advanced operator is that it provides decision-makers with a more comprehensive understanding of the situation. If we compare the results of this study to those of prior strategies, we can see that the approach proposed in this study is more thorough, more precise, and more concrete. As a result, this technique makes a significant contribution to the solution of real-world problems. Eventually, the suggested operator is put into practise in order to overcome the issues related to multi-attribute decision-making under the PF data environment. A numerical example has been used to show that the suggested method is valid, useful, and effective.

This paper focuses on t-norms which have some additive generators. We show that a t-norm T has some additive generators if and only if the fully ordered semigroup induced by Tp can be embedded into some fully ordered semigroup induced by either the product T or the Lukasiewicz t-norm Tp.

Ordered functional weighted averaging operators (OFWA operators) represent a generalization of OWA operators that are commonly used in decision making. It is shown that the class of OFWA operators is identical to the class of all intermediate functions, i.e. functions between min and max. The involved representation of OFWA operators determines a mapping between the set of all functions defined on the unit real interval and the family of all intermediate functions. The aim of this paper is to study some properties of this representation, in particular from a topological and algebraic point of view.

Decomposition integral with respect to a capacity (monotone measure) is a common framework for unifying some nonlinear integrals, such as the Choquet, the Shilkret, the PAN, the PC, and the concave integrals. The formula of decomposition integral concerning capacities depends on the distinguished decomposition system under some constraints on the sets being considered for each related integral. The M & ouml;bius representation of a monotone set function is a fundamental concept permitting the derivation of simple expressions of nonlinear integrals. In this paper, we propose M & ouml;bius representation of some decomposition integrals based on Lov & aacute;sz idea of an extension of pseudo-boolean functions for the decomposition systems to get different types of decomposition integrals. Then, we introduce a new type of integral to unify the Choquet, Shilkret, PAN, and PC integrals in terms of the M & ouml;bius transform. Consequently, we then propose the expressions of computing decomposition integrals concerning a 2-additive capacity. Finally, an illustrative example is used to demonstrate the applicability of the proposed results and simplicity of computing the 2-additive decomposition integrals.

The idea of decomposition integral, inspired by the concept of Lebesgue integral, is a common framework for unifying many nonlinear integrals, such as the Choquet, the Shilkret, the PAN, and the concave integrals. This framework concerns aggregation on a unipolar scale, and depends on the distinguished decomposition system under some constraints on the sets being considered for each related integral. The aim of this paper is to provide a general framework to deal with integrals concerning aggregation on unipolar and bipolar scales. To achieve this aim, we propose in this paper an extension of the idea of decomposition integral of the integrated function to be suitable for bipolar scales depending on the distinguished bipolar decomposition system under some constraints on the bipolar collections being considered for each related bipolar fuzzy integral. Then, we introduce some properties of bipolar decomposition integrals, including those establishing that our approach covers the Cumulative Prospect Theory (CPT) model and the integrals with respect to bipolar capacities. Finally, we conclude with certain directions on some additional findings related to the research.

Basic uncertain information is a recently introduced and significant type of uncertainty that proves particularly valuable in decision-making environments with inherent uncertainties. In this study, we propose the concept of uncertainty cognition merging, which effectively combines basic uncertain information granules with probability measures to generate new probability measures within the same probability space. Additionally, we present a degenerated method that merges basic uncertain information granules with unit intervals to create new subintervals. We introduce four distinct uncertainty cognition merging methods and thoroughly compare and analyze their respective properties, limitations, and advantages. To demonstrate the practical application potential of our proposals, we provide numerical examples alongside further mathematical results. ence information that can help them adjust their initial evaluation or prediction values. These external agencies often have access to statistical and probabilistic information based on history or big data, which generally gives them certain advantages such as appearing more objective and closer to reality and the market. However, there are also obvious drawbacks: these data are more general and may not be well-suited for specific decision-making scenarios; moreover, these data tend to be delayed, increasing the error rate of decisions. On the other hand, the core decision-making group within an organization is usually composed of internal experts or specially invited professionals who provide evaluations and predictions that are more specialized and relevant to the company's products. The downside of expert evaluations is that they tend to be subjective and with a significant amount of uncertainty. In summary, both internal evaluations and external consultations are important means of obtaining evaluation or prediction data. If internal experts assess that there is no uncertainty, decision-makers often tend to forgo seeking external consulting firms, even though these assessments may have some subjectivity, due to the high consultancy fees charged by such firms. However, in many cases, constrained by factors such as individual knowledge, experience, abilities, time and dedication of each expert, expert assessments carry significant uncertainties. This necessitates the reliance on statistical or probabilistic information provided by external consulting firms in order to make more informed assessments and decisions. The integration and merging of this external consulting information with the initial uncertain evaluations made by internal experts to obtain a more reasonable assessment result is the issue that will be discussed in this article.