Equivalence Relation Research Articles

Fuzzy Relative Homotopy and Fuzzy Weak Equivalence with Some Results

In this paper, first introduce my concept fuzzy homotopic and fuzzy homotopic relative and weproved that the relation fuzzy homotopic relative is a fuzzy equivalence relation. Secondly , weintroduce the concepts fuzzy homotopy equivalence, fuzzy fundamental group and fuzzy weak homotopy equivalence. We have proven some important theorems.

Uncertainty principles for short‐time free metaplectic transformation

This study devotes to extend Heisenberg's uncertainty inequalities in free metaplectic transformation (FMT) domains into short‐time free metaplectic transformation (STFMT) domains. We disclose an equivalence relation between spreads in time‐STFMT and time domains, as well as FMT‐STFMT and FMT domains. We use them to set up an inequality relation between the uncertainty product in time‐STFMT and FMT‐STFMT domains and that in time and FMT domains and an inequality relation between the uncertainty product in two FMT‐STFMT domains and that in two FMT domains. We deduce uncertainty inequalities of real‐valued functions and complex‐valued window functions for the STFMT and uncertainty inequalities of complex‐valued (window) functions for the orthogonal STFMT, the orthonormal STFMT, and the STFMT without the assumption of orthogonality, respectively. To formulate the attainable lower bounds, we also propose some novel uncertainty inequalities of complex‐valued (window) functions for the orthogonal FMT and the FMT without the assumption of orthogonality, respectively.

Rigid stabilizers and local prosolubility for boundary-transitive actions on tree

Abstract Let 𝐺 be a group acting 2-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where 𝐺 has both P ⁢ Γ ⁢ L 3 ⁢ ( 4 ) \mathrm{P}\Gamma\mathrm{L}_{3}(4) and P ⁢ Γ ⁢ L 3 ⁢ ( 5 ) \mathrm{P}\Gamma\mathrm{L}_{3}(5) as local actions. We show that, for each half-tree T a T_{a} , the local action of the rigid stabilizer of T a T_{a} at the root of T a T_{a} contains the soluble residual of the point stabilizer of the local action of 𝐺. In particular, 𝐺 is locally prosoluble if and only if its local actions have soluble point stabilizers; if 𝐺 is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in 𝐺. These results are partly inspired by Radu’s classification of groups acting boundary-2-transitively on trees with local action containing the alternating group, and partly based on the author’s recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks.

The L-Fuzzy Prime Filter Degrees on Lattices and Its Induced L-Fuzzy Convex Structure

The aim of this paper is to examine the L-fuzzy prime filter degrees on lattices and their induced L-fuzzy convex structure. Firstly, the notion of L-fuzzy prime filter degrees on lattices is established using the implication operator when L is a completely distributive lattice. Secondly, an equivalent characterization of L-fuzzy prime filter degrees on lattices is provided. The equivalence relation, through the definitions of reflexivity, symmetry, and transitivity, provides a method for partitioning subsets within a lattice that possesses the prime filter property. Finally, the L-fuzzy convex structure induced by the L-fuzzy prime filter degrees on lattices is examined. Simultaneously, the properties of L-fuzzy prime filter degrees on lattices in relation to images and preimages under homomorphic mappings are discussed.

Generalization of filter theory of EQ-algebras based on lattices

Abstract In this article, considering the notion of a lattice-valued set (${\mathcal{L}}$-set, for short) we introduce the notions of ${\mathcal{L}}$-prefilter and ${\mathcal{L}}$-filter in EQ-algebras. We provide several characterizations and equivalent conditions for these concepts and also characterize the ${\mathcal{L}}$-prefilter and ${\mathcal{L}}$-filter generated by an ${\mathcal{L}}$-set. Subsequently, we study the lattice structure of these filters and prove that in an $\ell $EQ-algebra, the lattice of ${\mathcal{L}}$-prefilters is a complete Brouwerian lattice, and hence it forms a Heyting lattice. In a residuated $\ell $EQ-algebra, we show that the lattice of ${\mathcal{L}}$-filters is also a Heyting lattice. They also form a semi-De Morgan algebra. Moreover, we demonstrate that the skeleton of an $\ell $EQ-algebra under appropriate operations forms a Boolean algebra. Furthermore, we introduce (relative) ${\mathcal{L}}$-congruences in EQ-algebras and investigate their properties. We also explore the relationships between ${\mathcal{L}}$-prefilters/${\mathcal{L}}$-filters and ${\mathcal{L}}$-congruences. We prove that ${\mathcal{L}}$-prefilters induce a relative ${\mathcal{L}}$-equivalence relation and ${\mathcal{L}}$-filters correspond to relative ${\mathcal{L}}$-congruences, and state and prove some isomorphism theorems.

Modular structure theory on Hom-Lie algebras

Abstract The aim of this paper is to transfer the restrictedness theory to Hom-Lie algebras. The concept of restricted Hom-Lie algebras, which is introduced in [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No. 4955], will be used in this paper. First, the existence and uniqueness of p-structures on a Hom-Lie algebra is studied. Then the definition of a restrictable Hom-Lie algebra is given and the equivalence relation between restrictable Hom-Lie algebras and restricted Hom-Lie algebras is constructed. Finally, the p-envelopes of a Hom-Lie algebra are defined and studied.

Strong ergodicity around countable products of countable equivalence relations

Strong ergodicity around countable products of countable equivalence relations

On I -Statistically Cauchy Sequences & the Class of I -Statistically Concurrent Sequences

Abstract In this article, we present the ideas of I-statistically Cauchy criteria and I ∗ -statistically Cauchy criteria, which are the generalizations of I-Cauchy and I ∗ -Cauchy criterion, respectively. To grasp the differences, we compare this I-statistically Cauchy criterion with a few other Cauchy criteria. Also, we investigate a few characteristics of I ∗ -statistically Cauchy sequences and I-statistically Cauchy sequences and demonstrate their equivalence under the condition that the ideal I satisfies the property (AP). Furthermore, a relation is defined on the set SX of all sequences in a metric space, which comes out to be an equivalence relation. Finally, we show that if two sequences belong to the same equivalence class, then either both of them are I-statistically Cauchy or none of them are.

ОТНОШЕНИЕ НА ЕКВИВАЛЕНТНОСТ НА ИЗБРАНИ ПОЛСКИ И БЪЛГАРСКИ ТЕРМИНИ В СЪДЕБНИ АКТОВЕ (ПО НАКАЗАТЕЛНИ ДЕЛА)

The article juxtaposes Polish and Bulgarian terms in terms of the names of courts, the names of court authorities, assistants, parties and other persons involved in the case in Polish and Bulgarian. The compilation of the juxtaposition made it possible to resolve what type of equivalence relations we are dealing with in the type of court records under study. To determine the type of equivalence relations between Polish and Bulgarian legal terms, I used the concept of denotative equivalence distinguished by Wolter Koller. Recognizing the types of equivalence relations is the starting point for considering the difficulties and translation strategies that the translator must undertake.

A fatigue life approach model of automotive parts based on stress equivalence relation

A fatigue life approach model of automotive parts based on stress equivalence relation

Feature selection technique on convolutional neural network – multilabel classification task

Automated text-based recommendation, an artificial intelligence development, finds application in document analysis like job resumes. The classification of job resumes poses challenges due to the ambiguity in categorizing multiple potential jobs in a single application file, termed multi-label classification, deep learning, particularly convolutional neural networks (CNN), offers flexibility in enhancing feature representations. Despite its robust learning capabilities, the black-box design of deep learning lacks interpretability and demands a substantial number of parameters, requiring significant computational resources. The primary challenge in multilabel learning is the ambiguity of labels not fully explained by traditional equivalence relations. To address this, the research employs feature selection techniques, specifically the Chi-square method. The goal is to reduce features in deep learning models while considering label relevance in multi-label text classification, easing computational workload while preserving model performance. Experimental tests, both with and without the Chi-square feature selection technique on the dataset, underscore its substantial impact on the classification model's ability. The conclusion emphasizes the influence of the Chi-square feature selection technique on performance and computational time. In summary, the research underscores the importance of balancing computational efficiency and model interpretability, especially in complex multi-label classification tasks like job applications.

Effective Value Analysis of Fuzzy Similarity Relation in HQSS for Efficient Granulation.

Hierarchical quotient space structure (HQSS), as a typical description of granular computing (GrC), focuses on hierarchically granulating fuzzy data and mining hidden knowledge. The key step of constructing HQSS is to transform the fuzzy similarity relation into fuzzy equivalence relation. However, on one hand, the transformation process has high time complexity. On the other hand, it is difficult to mine knowledge directly from fuzzy similarity relation due to its information redundancy, i.e., sparsity of effective information. Therefore, this article mainly focuses on proposing an efficient granulation approach for constructing HQSS by quickly extracting the effective value of fuzzy similarity relation. First, the effective value and effective position of fuzzy similarity relation are defined according to whether they could be retained in fuzzy equivalence relation. Second, the number and composition of effective values are presented to confirm that which elements are effective values. Based on these above theories, redundant information and sparse effective information in fuzzy similarity relation could be completely distinguished. Next, both isomorphism and similarity between two fuzzy similarity relations are researched based on the effective value. The isomorphism between two fuzzy equivalence relations is discussed based on the effective value. Then, the algorithm with low time complexity for extracting effective values of fuzzy similarity relation is introduced. On the basis, the algorithm for constructing HQSS is presented to realize efficient granulation of fuzzy data. The proposed algorithms could accurately extract effective information from the fuzzy similarity relation and construct the same HQSS with the fuzzy equivalence relation while greatly reducing the time complexity. Finally, relevant experiments on 15 UCI datasets, 3 UKB datasets, and 5 image datasets are shown and analyzed to verify the effectiveness and efficiency of the proposed algorithm.

Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups

Let TX be the full transformation monoid over a finite set X, and fix some a∈TX of rank r. The variant TXa has underlying set TX, and operation f⋆g=fag. We study the congruences of the subsemigroup P=Reg(TXa) consisting of all regular elements of TXa, and the lattice Cong(P) of all such congruences. Our main structure theorem ultimately decomposes Cong(P) as a specific subdirect product of Cong(Tr), and the full equivalence relation lattices of certain combinatorial systems of subsets and partitions. We use this to give an explicit classification of the congruences themselves, and we also give a formula for the height of the lattice.

Interval-Valued Multiobjective Programming Problems Based on Convex Cones

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients.

Reduced-order reconstruction of discrete grey forecasting model and its application

Discrete grey forecasting models based on an accumulative operator have been widely used in many practical fields. With the development of grey forecasting models, it is a problem to be solved to further analyze internal mechanisms and unify the structures. This paper aims to reconstruct the model from a perspective of sequence characteristics and simplify the modeling steps under the condition of ensuring the accuracy of the model. First, this paper analyzes dynamic sequence evolution hidden and mines relationship between the structure and original sequence features contained in discrete grey forecasting model. Then, the reconstruction is carried out to prove the equivalence and quantitative relation between reduced-order model and original model. Under order recursive estimation, new parameters are addressed. Finally, theoretical correctness is verified by large-scale numerical simulation. Moreover, the reduced-order model is applied for prediction on the peak of battery incremental capacity and capacity degradation. Results show the effectiveness and superior prediction performance of the reduced-order model, where MAPEs of grey forecasting models have controlled under 4%.

Three-way concept lattice from adjunctive positive and negative concepts

Three-way concept lattices (TCLs) have been widely explored due to their clear hierarchical structures, concise visual description and good interpretability. In contrast to classic formal contexts, lattice-valued fuzzy contexts exhibit great capability in describing and representing concepts with uncertainty. Different from conventional approaches to research of TCLs, this paper focuses on investigating the algebraic structure and properties of three-way concept lattice (TCL) stemmed from the positive concept lattice and negative concept lattice in a lattice-valued formal context. Several associated concept lattices such as the Cartesian product of positive concept lattice and negative lattice (i.e., pos-neg lattice), lattices induced from the partition of the pos-neg lattice, and their relationship are explored. Specifically, the isomorphism, embedding and order-preserving mappings between them are built. The quotient set of pos-neg lattice when being defined a specific equivalence relation on it is a complete lattice and each equivalence class is a lower semi-lattice. It is further declared that the structure of TCL is intrinsically and determined wholly by the pos-neg lattice. A practical application of the built theory of TCL is provided to sort alternatives in multi-criteria decision making.

Relating in the Wild: Toward an Analysis of Equivalence Relations Under More Naturalistic Conditions.

Since the first proposal 50years ago, numerous experiments have documented how arbitrarily related stimuli can become functionally interchangeable. These studies have sought to understand how different variables can moderate the probability of equivalence class formation. However, the well-established evidence regarding this phenomenon in experimental settings does not necessarily guarantee an understanding about how equivalence relations are produced in natural settings. In typical experiments, experimenters control critical variables to produce equivalence relations, such as, the requirement of proficiency with baseline relations, the number of opportunities to relate two or more stimuli, the efforts to promote stimulus control topography coherent with the experimenter-defined relations, etc. All these variables, however, are not controlled in our daily lives. The present article elucidates how some differences between experimental and natural settings can likely affect how the phenomenon of equivalence relations can occur in noncontrolled, naturalistic environments. Furthermore, we suggest new areas of research to promote the generalization of basic experimental data to contingencies in our daily lives.

Grey relations

PurposeThe goal of this article is to introduce the notion of a grey relation between grey sets using grey numbers.Design/methodology/approachThis study uses the grey number to create novel ideas of grey sets. We suggest several operations that can be performed on it, including the union, intersection, join, merge, and composition of grey relations. In addition, we present the definitions of reflexive, symmetric, and transitive grey relations and analyze certain characteristics associated with them. Furthermore, we formulate the concept of the grey equivalence relation, apply it to the study of the grey equivalence class over the grey relation, and go over some of its features.FindingsWe present new algebraic aspects of grey system theory by defining grey relations and then analyzing their characteristic features.Practical implicationsThis paper proposes a new theoretical direction for grey sets according to grey numbers, namely, grey relations. This paper proposes a new theoretical direction for grey sets according to grey numbers, namely, grey relations. As such, it can be applied to create rough approximations as well as congruences in algebras, topologies, and semigroups.Originality/valueThe presented notions are regarded as new algebraic approaches in grey system theory for the first time. Additionally, some fundamental operations on grey relations are also investigated. Consequently, different types of grey relations, such as reflexive, symmetric, and transitive relations, are discussed. Then, the grey equivalence class derived from the grey equivalence relation is demonstrated, and its properties are examined.

A Proven Translation from a UML State Machine Subset to Timed Automata

Although Unified Modeling Language (UML) state machines constitute a convenient modeling formalism that is widely used in many applications, the lack of formal semantics impedes carrying out automatic processing, such as formal verification. In this article, we aim to achieve a proven translation from a subset of UML state machines to timed automata. A generic abstract syntax is defined for state machines that allows us to specify state machines as a tree-like structure, explicitly illustrating the hierarchical relationships within the model. Based on this syntax, a formal asynchronous semantics for state machines and systems of state machines is established. Additionally, the semantics of timed automata is specified. Then, a translation relation from the considered set of state machines to timed automata is defined and a strong equivalence relation — namely, a timed bisimulation between the source and target models — is formally proven. The proof is carried out inductively while considering continuous (time) and discrete transitions separately. This proof allows us to demonstrate a strong similitude between these models.

A cost-sensitive decision model for efficient pooled testing in mass surveillance of infectious diseases like COVID-19

The COVID-19 pandemic has imposed significant challenges on global health, emphasizing the persistent threat of large-scale infectious diseases in the future. This study addresses the need to enhance pooled testing efficiency for large populations. The common approach in pooled testing involves consolidating multiple test samples into a single tube to efficiently detect positivity at a lower cost. However, what is the optimal number of samples to be grouped together in order to minimize costs? i.e. allocating ten individuals per group may not be the most cost-effective strategy. In response, this paper introduces the hierarchical quotient space, an extension of fuzzy equivalence relations, as a method to optimize group allocations. In this study, we propose a cost-sensitive multi-granularity intelligent decision model to further minimize testing costs. This model considers both testing and collection costs, aiming to achieve the lowest total cost through optimal grouping at a single layer. Building upon this foundation, two multi-granularity models are proposed, exploring hierarchical group optimization. The experimental simulations were conducted using MATLAB R2022a on a desktop with Intel i5-10500 CPU and 8G RAM, considering scenarios with a fixed number of individuals and fixed positive probability. The main findings from our simulations demonstrate that the proposed models significantly enhance the efficiency and reduce the overall costs associated with pooled testing. For example, testing costs were reduced by nearly half when the optimal grouping strategy was applied, compared to the traditional method of grouping ten individuals. Additionally, the multi-granularity approach further optimized the hierarchical groupings, leading to substantial cost savings and improved testing efficiency.