Fuzzy graphs have numerous applications in real life. Up till now, there have been many extensions of fuzzy graphs. For example, intuitionistic fuzzy graphs, Pythagorean fuzzy valued neutrosophic fuzzy graphs, bipolar fuzzy graphs, m-polar fuzzy graphs, and so on. In the present paper, we shall present a novel definition of lattice-valued fuzzy graphs by lattice implication operators 7-+, and prove that various fuzzy graphs mentioned above can be regarded as special lattice-valued fuzzy graphs. Moreover we shall present some characterizations of lattice-valued fuzzy graphs.

Based on a complete Heyting algebra L, we first propose the concept of L-quasi-convex spaces and construct an adjunction between the category of L-S0-quasi-convex spaces and the opposite category of complete L-ordered sets. Then we present the concept of weakly fuzzy algebraic lattices and prove that an L-quasi-convex structure endowed with the fuzzy inclusion order is precisely a weakly fuzzy algebraic lattice. Secondly, we introduce the notion of sobriety in L-quasi-convex spaces from the perspective of categorical equivalence, showing that the category of sober L-quasi-convex spaces is dually equivalent to that of weakly fuzzy algebraic lattices. Finally, we construct a monad on the category of L-S0-quasi-convex spaces and obtain that the EilenbergMo ore algebras of this monad are precisely sober L-quasi-convex spaces.

As two non-associative binary fuzzy logical operators, overlap and grouping functions have been integrated into rough set theory. However, research on covering-based variable precision fuzzy rough sets (CVPFRSs) from the perspective of overlap and grouping functions remains limited, leaving several gaps to be addressed. To fill this gap, we propose novel CVPFRS models based on overlap and grouping functions, referred to as (O, G)-CVPFRSs, and develop corresponding multi attribute decision-making (MADM) methods. First, by employing residual implications and coimplications derived from overlap and grouping functions, we construct four distinct (O, G)-CVPFRS models and systematically investigate their theoretical properties, with particular emphasis on their comparability property. Subsequently, building upon the traditional TOPSIS method, we propose two MADM methods grounded in the (O, G)-CVPFRS models. Finally, we validate the proposed methods through a numerical case study. Comparative analyses with benchmark approaches demonstrate the validity, reliability, and practical effectiveness of the proposed methods in material selection for bone grafting.

The primary objective of this paper is to study the properties of inverted perpendicular-closed sets and the largest T-2 inverted perpendicular-compactification of inverted perpendicular-convergence spaces. Firstly, we study some properties of inverted perpendicular-ultrafilters and introduce the concept of a inverted perpendicular-closed set and the concept of a inverted perpendicular-compact set, examining the relationship between them. Secondly, we present the notion of essentially inverted perpendicular-compact inverted perpendicular-convergence spaces and explore the necessary and sufficient conditions for a inverted perpendicular-convergence space to have the largest T-2 inverted perpendicular-compactification. Finally, we construct the Richardson inverted perpendicular-compactification of a inverted perpendicular-convergence space and identify the necessary and sufficient conditions for the Richardson inverted perpendicular-compactification to be the largest T-2 inverted perpendicular-compactification within the framework of Kent inverted perpendicular-convergence spaces. (c) 2025 Published by Elsevier B.V.

In this paper, we introduce two types of Levitin-Polyak well-posedness for split quasi-equilibrium problems. We establish different characterizations of these well-posedness notions with and without gap functions for split quasi-equilibrium problems. Furthermore, we provide equivalence between the well-posedness of constrained optimization problems and that of split quasi-equilibrium problems using gap function techniques. By analyzing the upper semicontinuity of approximate solution sets, we derive necessary and/or sufficient conditions for type I Levitin-Polyak well-posedness. Numerical examples are provided to validate our theoretical findings.

Quotient space theory provides an effective mechanism for transforming fine-grained problem representations into coarse-grained ones, which is beneficial for knowledge reduction and decision analysis in complex systems. However, classical quotient space models rely heavily on equivalence relations, making them unsuitable for practical scenarios where data exhibit vagueness, uncertainty, and indistinct boundaries. In addition, existing fuzzy rough set-based feature selection methods often suffer from high computational cost, especially in fine-grained information systems. To address these issues, this paper proposes a generalized quotient space framework based on fuzzy neighborhood operators. Within this framework, a fuzzy quotient mapping is constructed, and a quotient fuzzy /3-covering approximation space (/3-QFCAS) is established. Based on /3-QFCAS, a fuzzy /3-covering rough set model is developed, and its key properties are systematically investigated. Moreover, a forward feature selection algorithm is designed in the quotient fuzzy /3-covering decision space (/3-QFCDS) to improve computational efficiency while preserving decision performance. Experimental results demonstrate that the proposed method achieves competitive accuracy, indicating its effectiveness for feature selection in uncertain and complex decision-making environments.

In image processing, denoising can provide high-quality data for subsequent processing. Current denoising methods can be categorized into three types: learningbased methods, model-based methods, and filtering-based methods. Although the first two methods excel at handling complex noise, traditional filtering methods remain advantageous in scenarios requiring high real-time performance and limited computational resources. In this paper, an adaptive fuzzy nonlinear filtering algorithm (NCFR algorithm) based on neighborhood-controlled fuzzy rough set model is proposed to effectively remove salt-and-pepper noise from images. The proposed NCFR algorithm introduces fuzzy membership functions to address the uncertainties and fuzziness between image pixels, and utilizes the noise-free pixels within an adaptive neighborhood to restore noisy pixels, thereby efficiently reducing noise while preserving image details. Experimental results demonstrate that the NCFR algorithm significantly improves image restoration quality under various noise densities. In particular, at high noise densities exceeding 95%, the algorithm shows superior performance in terms of PSNR and SSIM indices, better preserving structural characteristics and texture details of the images.

Based on a completely distributive lattice L, we propose a degree approach to L-fuzzy ordered subsemigroups of an ordered semigroup. Firstly, we introduce the concept of L-fuzzy ordered subsemigroup degree function with respect to an ordered semigroup, which can be used to describe the degree to which an L-fuzzy subset of the ordered semigroup becomes an L-fuzzy ordered subsemigroup. Secondly, we use four kinds of cut sets depending on L to characterize the L-fuzzy ordered subsemigroup degree function. Finally, we provide a natural way to construct an L-fuzzy convex structure on an ordered semigroup via the L-fuzzy ordered subsemigroup degree function, and show that the homomorphism between two ordered semigroups is an L-fuzzy convexity-preserving mapping and the monohomomorphism is an L-fuzzy convex-to-convex mapping between the resulting L-fuzzy convex spaces.

This paper develops a unified approach to the completion of T-filter spaces. First, a novel equivalence relation on a T-filter structure is introduced, which serves as the foundation for defining T-pre-Cauchy spaces. This equivalence relation establishes the connection between Tfilter spaces and T-convergence spaces, and provides a rigorous basis for defining completeness in T-filter spaces. On this basis, an equivalence-embedding completion of a T-filter space is constructed, together with corresponding extension theorems. Subsequently, the completion method is applied to both T-pre-Cauchy and T-Cauchy spaces. Moreover, alternative completion methods for these spaces are introduced, and a detailed comparison of their interrelations is carried out. In particular, the finest T1 equivalence-embedding completion is characterized in the setting of T-pre-Cauchy spaces.

In this paper, with a frame as the truth value table, we propose the concepts of fuzzy F-closure spaces and fuzzy IG-closure spaces. Based on these concepts, we provide representations of fuzzy algebraic dcpos and fuzzy domains, respectively. Furthermore, we introduce the notion of fuzzy F-relations, which accurately represent fuzzy Scott continuous maps between fuzzy algebraic dcpos. Consequently, we establish a categorical equivalence between fuzzy F-closure spaces and fuzzy algebraic dcpos. Moreover, we introduce the concept of approximable L-relations and demonstrate that the category of fuzzy IG-closure spaces is equivalent to that of fuzzy domains.

Based on a complete residuated lattice L, we show that the category of L-convex spaces is not extensional and is closed under the formation of finite products of quotient maps. Then we propose the concept of (preconcave, concave) L-convergence spaces via L-co-Scott closed sets and prove that the category of concave L-convergence spaces is isomorphic to that of L-concave spaces. Finally, we investigate the categorical properties of L-convergence spaces and show that it is extensional and closed under the formation of finite products of quotient maps.

Based on a complete residuated lattice L, we combine the lattice-valued coarse structures and group operations to propose the concept of L-fuzzifying coarse groups. Then we introduce the notion of L-fuzzifying group ideals and establish its one-to-one correspondence with L-fuzzifying coarse groups. Specifically, we examine how L-fuzzifying coarse structures align with the algebraic structures of the supporting group. Finally, we use L-fuzzifying group ideals to characterize a fuzzy coarse equivalence between L-fuzzifying coarse groups, presenting some results derived from the kernel of the group homomorphism.

Ternary fuzzy relations, and fuzzy betweenness relations in particular, are witnessing increasing attention in recent years. A key reason is that axiomatic properties of ternary fuzzy relations seem to be ideally suited to capture geometric characteristics of the abstract notion of betweenness. In this paper, we introduce several new properties of ternary fuzzy relations, including the Peano property, the Pasch property and the sand-glass property, that can be qualified as geometric properties. We investigate their interrelationships as well as their connections with various types of fuzzy betweenness relations. Additionally, in the context of our study of the Pasch property and the sand-glass property, we introduce the convexity property of ternary fuzzy relations by taking inspiration from the solid theoretical basis of the theory of fuzzy convex structures.

Matroid theory provides a broad theoretical framework with a wide range of applications. On this basis, fuzzy matroids have also undergone significant development based on lattice theory, fuzzy topology, fuzzy convexity, etc. In this paper, we select an overlap function to model conjunction to provide two fuzzification approaches to matroids. Taking an arbitrary overlap function (R), we introduce (R)-fuzzifying matroids and (R)-fuzzy matroids and study their relations with classical matroids, respectively. Then we explore the relations between (R)-fuzzifying matroids and (closed, perfect) (R)-fuzzy matroids. These findings not only provide a theoretical application of overlap functions but also elucidate the connections between different fuzzifications of matroids.

For a completely distributive lattice V, a novel class of lattice-valued Scott open sets, referred to as Scott open V-sets, is introduced on the powerset. These sets are utilized to construct a monad over the category of sets, termed the Scott open V-set monad. It is demonstrated that the category of Eilenberg-Moore algebras for the Scott open V-set monad is isomorphic to that of algebraic V-modules, and the category of Kleisli monoids with respect to this monad is isomorphic not only to the category of algebraic V-closure spaces but also to that of lax algebras for the finite powerset monad.

Bounded trellises, also known as weakly associative lattices, offer a new structural basis for exploring aggregation functions. This study introduces three distinct approaches to construct uninorms on a bounded trellis, each based on a t-norm defined within a subinterval of the trellis. Through several examples, we demonstrate that the proposed constructions differ essentially from existing ones. On this foundation, the framework of aggregation operators on bounded trellises is further extended, with a focus on the theoretical development of uninorms.

Based on a complete residuated lattice L, we propose a new approach to fuzzification of coarse structures. Firstly, we introduce the concepts of L-entourages and L-(quasi, semi)-coarse structures and study the maps between L-coarse spaces. Secondly, we provide some examples of L-coarse structures from the aspects of L-metrics, L-relations and L-hyperstructures. Thirdly, we introduce the notion of L-ball structures to characterize L-coarse structures. Finally, we establish the relationship among L-entourage spaces, L-quasi-coarse spaces, L-semi-coarse spaces and L-coarse spaces in a categorical viewpoint.

The integration of three-way decision (3WD) into multiple attribute decision-making (MADM) problems has emerged as a pivotal research area. 3WD can effectively manage the inherent uncertainty within the decision-making process. Additionally, it offers a semantic interpretation of the outcomes. In this paper, we introduce two innovative 3WD-MADM approaches, with a focus on granule selection and the handling of multi-type information in the framework of three-way decisions. Firstly, we construct maximal consistent blocks (MCBs)-based pessimistic and optimistic probabilistic rough fuzzy set (RFS) models and investigate their properties to ascertain their efficacy and reliability in decision-making contexts. Then, we define relative loss functions associated with "good state" and "bad state" scenarios. Building on this, we introduce four types of 3WDs based on our newly proposed optimistic and pessimistic probabilistic RFSs. Furthermore, we integrate the 3WDs information from both scenarios to formulate optimistic and pessimistic 3WD-MADM approaches, handling both single-valued fuzzy and intuitionistic fuzzy information. Finally, we contrast our proposed methodologies with the current MADM methods, and demonstrate their validity, significance and generalization ability.

Ultrafilters serve as an important tool for studying compactness and Choquet convergence structures in classical convergence spaces. In the framework of T-convergence spaces, we provide three characterizations of T-ultrafilters and consider their applications from three aspects. Firstly, we use T-ultrafilters to study the T-compactness of a T-convergence space, including the Tychonoff theorem and the relationships between the compactness of a classical convergence space and its induced T-convergence space. Secondly, we use T-ultrafilters to construct the one- point T2 T-compactification of a T-convergence space and present the necessary and sufficient conditions for one-point T2 T-compactification to be the smallest. Finally, we employ T-ultrafilters to define Choquet T-convergence spaces and investigate their function spaces as well as their relationships with other types of T-convergence spaces.

The primary objective of this paper is to study the properties of ⊤-closed sets and the largest T2 ⊤-compactification of ⊤-convergence spaces. Firstly, we study some properties of ⊤-ultrafilters and introduce the concept of a ⊤-closed set and the concept of a ⊤-compact set, examining the relationship between them. Secondly, we present the notion of essentially ⊤-compact ⊤-convergence spaces and explore the necessary and sufficient conditions for a ⊤-convergence space to have a largest T2 ⊤-compactification. Finally, we construct the Richardson ⊤-compactification of a ⊤-convergence space and identified the necessary and sufficient conditions for the Richardson ⊤-compactification to be the largest T2 ⊤-compactification within the framework of Kent ⊤-convergence spaces.