Partition-based clustering is widely adopted for its simplicity and efficiency. However, it fails to capture non-convex, manifold-based, or linearly inseparable clusters, which commonly appear in real-world data, such as biological sequences, social networks, and medical images. This limitation severely restricts its application in recognizing clusters of arbitrary shapes. To address this challenge, we propose TC-KMeans, a$k$-means method integrated with transitive closure. Its core innovation lies in the incorporation of transitive closure into distance functions: instead of relying solely on direct pairwise distances, TC-KMeans establishes sample associations by propagating relevance through transitive closure. This design allows it to treat highly correlated samples as a cohesive whole, even when the direct distance between data points is large. These improvements overcome the convex structure limitation of traditional partition-based methods, enabling TC-KMeans to detect arbitrary -shaped clusters while enhancing both clustering accuracy and completeness. To validate the proposed algorithm, we used Purity, AMI, ARI, FMI, and computation time as evaluation metrics, and conducted tests comparing it with 11 comparative algorithms on 20 synthetic datasets and 10 real-world datasets of different dimensions. Experimental results show that on synthetic non-convex datasets, TC-K-Means achieves a value of 1.0in key metrics for most datasets and has a shorter computation time than other algorithms; on real-world datasets, among the 12 algorithms, its average ranking in all metrics ranks first, outperforming traditional partition-based methods and density-based methods. In conclusion, TC-KMeans provides a practical solution for clustering complex non-convex data and expands the application scope of partition-based clustering algorithms.

Abstract In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces, and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.

Inspired by reflection transformations and rotation transformations in plane geometry and dictionary orders in lattice theory, in this article, we explore, through exposing the possible motivation of definition of a defined linear order ≤ on R (the set of all interval numbers), how to find an easy-to-understand way to extend the ordinary linear order on R (the set of all real numbers) to a linear order on R. Theory and application aspects are also studied. For an arbitrary cardinal number κ (κ ≠ 0), we define seven operations and the point-wise order induced by ≤ on Rκ(the set of all sequences indexed by κ and consisting of interval numbers). It is found that ( R,\≤), much like R (with the ordinary linear order), has some desirable properties; particularly, ≤ can be used to describe the corrected degree of possibility of an interval number is smaller than another. Generalizing the discovery process of ≤, we provide an understandable way to extend the ordinary linear order on R to a linear order on R (even on the plane R2), which meets some specific requirement, and we give a clear description on the relation between the admissible orders and those we propose. Those urge us to take a look at the possible applications of ≤ (a delegation of linear orders on R). Precisely, to propose an ordering method (based on this linear order and the matched metric and weighted averaging aggregation operator), which can be used to deal with very complicated generalized interval-valued preference hesitant relations in group decision-making problems.

This paper studies the fuzzy Scott topology on dcpos with a *-continuous semigroup (L, *) as the truth value table. It is shown that the fuzzy Scott topological space on a continuous dcpo is an ιL-sober space. The fuzzy Scott topology is completely distributive iff L is completely distributive and the underlying dcpo is continuous. For (L, *) being an integral quantale, semantics of L-possibility of computations is studied by means of a duality.

For a frame L as the truth value table, we apply fuzzy domain theory for the study of injective objects in the category of stratified L-T0spaces. We show that every fuzzy continuous lattice equipped with the fuzzy Scott topology is an injective stratified L-T0space, and conversely, the specialization L-ordered set of an injective stratified L-T0space is a fuzzy continuous lattice. These two transformations form a categorical isomorphism between injective stratified L-T0spaces and fuzzy continuous lattices.