Abstract Let $\textbf{Lat}_{\lor }$ denote the category of lattices with morphisms preserving supremum of non-empty finite subsets. We prove the following: (1) There exists a distributive law of the ordered-ideal monad over the ordered-filter monad on $\textbf{Lat}_{\lor }$. (2) The category ${\textbf{Cnt}}^{\mathbb{co}}$ of cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is strictly monadic over $\textbf{Lat}_{\lor }$. (3) The category ${\textbf{MCnt}}^{\mathbb{co}}$ of meet-continuous and cocontinuous lattices with morphisms preserving arbitrary non-empty sups and filtered infs is isomorphic to a full subcategory of ${\textbf{Lat}_{\lor }}^{\mathbb{FilId}}$, consisting of $\mathbb{FilId}$-algebras and $\mathbb{FilId}$-homomorphisms on ${\textbf{Lat}_{\lor }}$. For bounded-complete posets, analogous Eilenberg–Moore algebras provide a categorical representation. Finally, using the concept of bounded down-sets, we derive an algebraic characterization of ${{\textbf{Cnt}}}^{\mathrm{co}}$.

Within the framework of fuzzy algebras with fuzzy equality and a complete lattice for membership values, this paper introduces a novel fuzzified version of classical Green's relations on E -fuzzy semigroups, referred to as E -fuzzy Green's relations. We reveal several properties of E -fuzzy semigroups, including the properties of fuzzy unit elements within E -fuzzy cancellative semigroups. Additionally, we derive equivalent forms of these newly defined fuzzy relations and examine the properties of their associated cut-quotient structures in certain E -fuzzy semigroups. The E -fuzzy Green's relations on E -fuzzy semigroups are found to be compatible fuzzy equivalence relations (fuzzy equalities) on some E -fuzzy bands. Furthermore, we present two significant decomposition theorems related to the cut-quotient structures over fuzzy Green's relations, demonstrating that these structures have additional algebraic properties compared with the quotient structures over ordinary fuzzy equality E μ .

Abstract Domains exhibit a variety of different aspects, some are order theoretical, some are topological, some belong to topological algebra. In this paper, we introduce two kinds of congruence relations on domains: I-congruence relation and II-congruence relation on domains. We obtain that there is a bijection from the set of all kernel operators of domain $P$ preserving directed sups onto the set of all I-congruence relations on $P$ which exclude $P\times P$ . There is also a bijection from the set of all closure operators of domain $P$ preserving directed sups onto the set of all II-congruence relations on $P$ which exclude $P\times P$ . Furthermore, between two domains, we propose a new homomorphism called I-homomorphism and II-homomorphism, respectively. We conclude that the kernels of I-homomorphisms and II-homomorphisms between domains are I-congruence relations and II-congruence relations on domains, respectively. Therefore, we obtain the I-homomorphism and I-isomorphism theorems, as well as II-homomorphism and II-isomorphism theorems for domains. Besides, we give a positive answer to an open problem on homomorphisms and quotients of continuous semilattices posed by G. Gierz, et al.

Feature selection can select important features to address dimensional curses. Subspace learning, a widely used dimensionality reduction method, can project the original data into a low-dimensional space. However, the low-dimensional representation is often transformed back into the original space, resulting in information loss. In addition, gate function-based methods in Takagi–Sugeno–Kang fuzzy system (TSK-FS) are commonly less discrimination. To address these issues, this article proposes a novel feature selection method that integrates subspace learning with TSK-FS. Specifically, a projection matrix is used to fit the intrinsic low-dimensional representation. Subsequently, the low-dimensional representation is fed to TSK-FS to measure its availability. The firing strength is slacked so that TSK-FS is not limited by numerical underflow. Finally, the$\ell _{2,1}$-norm is introduced to select significant features and the connection to related works is discussed. The proposed method is evaluated against six state-of-the-art methods on 17 datasets, and the results demonstrate the superiority of the proposed method.

In the era of Big Data, the integration and fusion of heterogeneous information have become essential for addressing complex decision-making challenges in large-scale group decision-making (LSGDM), where expanding scale and diverse participant behaviors increasingly demand advanced techniques to enhance efficiency, fairness, and accuracy. To meet these demands, this article proposes a novel framework that leverages information-fusion principles to optimize group decision-making processes. First, an enhanced fuzzy C-means algorithm, termed Trust-based regularized fuzzy C-means, is introduced. By incorporating a regularization term derived from trust relationships, it improves clustering precision and enables finer segmentation of decision-makers (DMs), thus laying solid groundwork for consensus. Building on this, we develop an optimization model integrating three-way decision theory to resolve asynchronous differences among experts. The model uses dynamic trust updates to guide DMs from the negative region toward those in the positive region, with special emphasis on rapid consensus in emergencies. Furthermore, the study, for the first time, systematically investigates manipulative and herd behaviors within large datasets; adopting a cautious strategy to mitigate their impact on fairness and effectiveness, it embeds pseudotrust identification into the framework to further refine the decision environment. Extensive case studies and comparative experiments demonstrate that the proposed method significantly improves decision-making efficiency, fairness, and accuracy, offering new perspectives and practical tools for managing complex information and optimizing group behaviors in LSGDM.

Abstract Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson’s $\mathbf{K}$ -ification. A subcategory of $\mathbf{TOP}_{\mathbf{0}}$ is called of type $\mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type $\mathrm K$ in the sense of Keimel and Lawson. Each such category induces a canonical monad $\mathcal K$ on the category $\mathbf{DCPO}$ of dcpos and Scott-continuous maps, which is called the order- $\mathbf{K}$ -ification monad in this paper. First, for each category of type $\mathrm{K}^{*}$ , we characterize the algebras of the corresponding monad $\mathcal K$ as k-complete posets and algebraic homomorphisms as k-continuous maps, from which we obtain that the order- $\mathbf{K}$ -ification monad gives the free k-complete poset construction over the category $\mathbf{POS}_{\mathbf{d}}$ of posets and Scott-continuous maps. In addition, we show that all k-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-K-ification monad and conclude with the fact that each order-K-ification monad is always commutative.

Abstract The study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $\Sigma P\times \Sigma Q$ coincides with the Scott topology of the product poset $P\times Q$ if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and $\Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.

Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

Real-time lower limbs motion or gait measurement is an important part in human-robotic interaction for the control of robotic walkers and rehabilitation devices. Laser range finder or infrared sensor that is mounted on the device has been widely used in applications. Although these sensors can provide accurate horizontal motion information of lower limbs during human walking, it is still difficult to measure the angular motion of lower limbs due to their functional principles. Using inertial measurement units (IMU) can measure the angular motion of lower limbs, but it requires a large amount of IMU units for measurements of all lower limb segments. In this study, a novel method is developed for real-time monitoring lower limbs (shanks and thighs) motion in human walking using just two shank-mounted IMUs. A pose prediction model based on multiple linear regression and Kalman filter is proposed. The root-mean-square error (RMSE) of the thigh orientation and the knee joint angle estimation in sagittal plane are 6.1 ± 1.3 and 6.8 ± 1.4 degs, respectively. The RMSE of the ankle, knee, and hip position estimation are 4.2 ± 1.3, 4.2 ± 1.1 and 3.5 ± 0.9 cm, respectively.

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

Inertial measurement units (IMU) have been used for gait analysis in many clinical studies, as a more convenient, low cost and less restricted alternative to the laboratory-based motion capture systems or instrumented walkways. Spatial-temporal gait parameters such as gait cycle duration and stride length calculated from the IMUs were often used in these studies for evaluating the impaired gait. However, the spatial-temporal information provided by IMUs is limited, and sometime suffers incomplete and less effective evaluation. In this study, we develop a novel IMU-based method for clinical gait evaluation. Nine gait variables including three spatial-temporal parameters and six kinematic parameters are extracted from two shank-mounted IMUs for quantifying patient's gait deviations. Based on those parameters, an IMU-based gait normalcy index (INI) is derived to evaluate the overall gait performance. Eight inpatient subjects with gait impairments caused by n-hexane neuropathy and ten healthy subjects were recruited. The proposed gait variables and INI were examined on the inpatients at three to five time instants during the rehabilitation process until being discharged. A comparison with healthy subjects and statistical analysis for the changes of gait variables and INI demonstrated that the proposed new set of gait variables and INI can provide adequate and effective information for quantifying gait abnormalities, and help understanding the progress of gait and effectiveness of therapy during rehabilitation process.

Abstract Sobriety, well-filteredness, and monotone convergence are three of the most important properties of topological spaces extensively studied in domain theory. Some other weak forms of sobriety and well-filteredness have also been investigated by some authors. In this paper, we introduce the notion of Θ-fine spaces, which provides a unified approach to such properties. In addition, this general approach leads to the definitions of some new topological properties.

Abstract We build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.

Abstract The notion of an m-algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m-approximable concept on each context. The m-approximable concept lattice derived from the notion is an m-algebraic lattice, and conversely, every m-algebraic lattice is isomorphic to an m-approximable concept lattice of some context. Morphisms on m-algebraic lattices and those on contexts are provided, called m-continuous functions and m-approximable morphisms, respectively. We establish a categorical equivalence between LATm, the category of m-algebraic lattices and m-continuous functions, and CXTm, the category of contexts and mapproximable morphisms.We prove that LATm is cartesian closed whenevermis regular and m > 2. By the equivalence of LATm and CXTm, we obtain that CXTm is also cartesian closed under same circumstances. The notions of a concept, an approximable concept, and a weak approximable concept are showed to be special cases of that of an m-approximable concept.

In this paper, we tailor-make new approximation operators inspired by rough set theory and specially suited for domain theory. Our approximation operators offer a fresh perspective to existing concepts and results in domain theory, but also reveal ways to establishing novel domain-theoretic results. For instance, (1) the well-known interpolation property of the way-below relation on a continuous poset is equivalent to the idempotence of a certain set-operator; (2) the continuity of a poset can be characterized by the coincidence of the Scott closure operator and the upper approximation operator induced by the way below relation; (3) meet-continuity can be established from a certain property of the topological closure operator. Additionally, we show how, to each approximating relation, an associated order-compatible topology can be defined in such a way that for the case of a continuous poset the topology associated to the way-below relation is exactly the Scott topology. A preliminary investigation is carried out on this new topology.

Recently, Rusu and Ciobanu established that for a continuous domainL, a subsetBofLis a basis if and only ifBis dense with respect to thed-topology, called the density topology, onL. In situations where directed completeness fails, Erné has proposed in 1991 an alternative definition of continuity calleds2-continuity which remedied the lack of stability of continuity under the classical Dedekind–MacNeille completion. In this paper, we show how the ‘Rusu–Ciobanu’ type of characterization can be formulated and established over the class ofs2-continuous posets with appropriate modifications. Although we obtain more properties of essential topologies and density topologies ons2-continuous posets, respectively.

We introduce a new concept of continuity of posets, called θ-continuity. Topological characterizations of θ-continuous posets are put forward. We also present two types of dcpo-completion of posets which are Dθ-completion and Ds2-completion. Connections between these notions of continuity and dcpo-completions of posets are investigated. The main results are (1) a poset P is θ-continuous iff its θ-topology lattice is completely distributive iff it is a quasi θ-continuous and meet θ-continuous poset iff its Dθ-completion is a domain; (2) the Dθ-completion of a poset B is isomorphic to a domain L iff B is a θ-embedded basis of L; (3) if a poset P is θ-continuous, then the Dθ-completion Dθ(P) is isomorphic to the round ideal completion RI(P, ≪θ).

In this paper, we introduce the notion of consistent F-augmented contexts by adding a special family of finite subsets into the structure of a formal context, which essentially establishes the basis of the representation of general algebraic domains. In particular, we investigate the association rule systems which are derived from the consistent F-augmented contexts and propose the notion of formal association rule systems. By the notion of antecedent connections, we obtain the equivalence between the category of formal association rule systems and that of algebraic domains, which demonstrates that the proposed notion of formal association rule systems provides a concrete approach to representing algebraic domains.

Abstract Rough set theory is a useful tool for dealing with imprecise knowledge. One of the advantages of rough set theory is the fact that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. This paper explores knowledge structures in a knowledge base. Knowledge structures in a knowledge base are firstly described by means of set vectors and relationships between knowledge structures divided into four classes. Then, properties of knowledge structures are discussed. Finally, group, lattice, mapping, and soft characterizations of knowledge structures are given.

The notion of homomorphism, as an important tool for studying the relationship between information systems, has attracted a great deal of attention in recent years, and the authors tend to pay their attention to static information systems in the exis

The introduction of information system homomorphisms has made a substantial contribution to attribute reduction. However, the efforts made on homomorphisms are far from sufficient. This paper further investigates homomorphisms between covering approximation spaces. First, we introduce the concepts of upper and lower homomorphisms as well as homomorphisms in order to study the relationship between covering approximation spaces. Then we present the notions of covering approximation subspaces and product spaces. We also compress covering approximation spaces and covering information systems with the aim of attribute reduction. Afterwards, by utilizing the compressions of the original spaces and systems we compress the dynamic covering approximation spaces and dynamic covering information systems. Several illustrative examples are employed to demonstrate that the homomorphisms provide an effective approach for compressing covering approximation spaces and covering information systems.

We studied the feasibility of estimating walking speed using two shank-mounted accelerometers. Our approach took advantage of the inverted pendulum-like behavior of the stance leg during walking to identify a new method for dividing up walking into individual stride cycles and estimating the initial conditions for the direct integration of the accelerometer signals. To test its accuracy, we compared speed to known values during treadmill walking. The speed estimation method worked well across treadmill speeds yielding a root mean square speed estimation error of only 8%. This accuracy is comparable to that achieved from shank-mounted inertial measurement unit, providing a robust and low-cost alternative in using accelerometer for walking speed estimation. Shank-mounted accelerometer may be of great benefit for estimating speed in walking for the embedded control of knee-mounted devices such as prostheses and energy harvesters.

A biomechanical energy harvester is presented that generates electricity during human walking. The key feature of this device is that the power generation adds only a minimal extra effort to the user. The knee-mounted devices accomplish this by selectively engaging power generation at the end of the swing phase when knee flexor muscles act to brake knee motion. Analogous to regenerative braking in hybrid cars, the device assists deceleration of each leg within each stride while generating electrical power. We developed a control system to engage/disengage power generation based on the measured knee kinematics during a gait cycle. Experimental results show that generative braking generated 4.8 plusmn 0.8 W of electrical power with a minimal increase in metabolic cost.

In the paper, we introduce the notion of approximable concept in fuzzy setting, and show that approximable concept lattices represent algebraic completely lattice L-ordered sets in the sense of R. Bělohlávek,.

This paper explores a cooperative manipulation method for orienting and translating convex objects in the plane. The manipulation task is performed by two agents using the concept of "virtual fence". During the manipulation, each agent makes a point contact with the object, and two agents push together along a straight-line. One advantage of a virtual fence over physical fence is that virtual fence can manipulate non-polygonal parts, and reduce the position and orientation uncertainties simultaneously. Moreover, the coordination between agents increases the flexibility of the manipulation system. We first identify two manipulation primitives, equilibrium and non-equilibrium pushes, and characterize the motion of the object under these two pushing actions. Then, we study the planning problem in the framework of switched system, and develop a fully analytical solution to the planning problem. Finally, manipulation examples and experiments are provided to demonstrate the proposed manipulation method

In this paper, neural networks are trained to compute the forward kinematics of spherical parallel manipulator (PM) for laparoscopic surgery application. Instead of solving a set of nonlinear equations for the forward kinematics, neural networks are used to map the input angles of revolute joints to the orientation of the manipulator. The training data are obtained from inverse kinematic relationships and measured from the experimental prototype model of the manipulator. Levenberg-Marquardt algorithm is used to train the neural networks, which leads to the fast convergence of the networks. The trained neural network model of forward kinematics are used in the real time interface between the graphical model and a haptic device for the laparoscopes surgery training application. Simulation and experiments are carried out to verify the performance of the proposed method.

This paper explores the unconstrained dynamic manipulation for parts on a plane using a one joint manipulator. The goal of dynamic manipulation is to pose an object from an initial to a desired goal configuration on a frictional supporting surface. The overall manipulation process is decomposed into two conjunct phases, namely, acceleration of the object and free sliding of the object. We study the acceleration of an object with a one joint manipulator, and develop a planning method by solving a free boundary value problem in this paper. Experimental results are presented to demonstrate the proposed manipulation approach.

In this paper, a novel numerical approach is proposed to solve the initial velocities of the free sliding object for given initial and final configurations. To find the desired initial velocities for free sliding objects is a key step for implementing dynamic manipulation. In order to plan the initial velocities, the motion of free sliding objects is modeled as a set of 6 first order differential equations, and the planning problem is formulated as a free boundary value problem (FBVP). Through a simple transformation, the FBVP is reduced to a standard Two-point boundary value (TPBV) problem. Quasi-Newton based optimization procedures are utilized to solve the planning problem. Unlike existing approaches, the proposed method does not require qualitative motion characteristics, thus it can be used for objects with general shape and arbitrary pressure distribution. Simulation results on polygonal objects with three to five vertices are used to demonstrate the planning method.

This paper proposes a novel planning method for multi-agent dynamic manipulation on a plane. The objective of planning is to find optimal forces exerted on the object by agents with which the object can follow a given trajectory. The main contributions of the proposed approach is: First, through integrating of noncooperative game and neural-net approximation, the planner can deal with unknown pressure distribution effectively. Second, by introducing cooperative game between agents, the forces exerted by agents distributed optimally. Based on the dynamic model of the pushed object, the planing problem is solved in two levels hierarchically. In the lower control level, generalized force inputs are designed by using minimax technique to achieve the tracking performance. In the coordination level, cooperative game is formulated between agents to distribute the generalized force, and the objective of the game is to minimize the worst case interaction force between agents and object. Simulations are carried out for the three-agent cooperative manipulation, results demonstrate the effectiveness of the proposed planning method.

This paper addresses the planning problem for multiagent dynamic manipulation in the plane. The objective of planning is to design the forces exerted on the object by agents with which the object can follow a given trajectory in spite of the uncertainty on pressure distribution. The main novelty of the proposed approach is the integration of noncooperative and cooperative games between agents in an hierarchical manner. Based on a dynamic model of the pushed object, the coordination problem is solved in two levels. In the lower control level, a fictitious force controller is designed by using a minimax technique to achieve the tracking performance. The design procedure is divided into two steps. First, a linear nominal controller is designed via full-state linearization with desired eigenvalues assignment. Next, a minimax control scheme is specified to optimally attenuate the worst-case effect of the uncertainty due to pressure distribution and achieve a minimax tracking performance. In the coordination level, a cooperative game is formulated between agents to distribute the fictitious force, and the objective of the game is to minimize the worst-case interaction force between agents and the object. Simulations are carried out for two-agent and three-agent manipulations, results demonstrate the effectiveness of the planning method.

In this paper, a stable fuzzy direct control scheme is presented for a class of interconnected nonlinear systems with unknown nonlinear subsystems and unknown nonlinear interconnections. In this control algorithm, fuzzy logic systems are employed to approximate the optimal controllers, which are designed on the assumption that all dynamics for each subsystem are known; then the fuzzy controllers and adaptation mechanisms for each subsystem depend only on local measurements to provide asymptotic tracking of a reference trajectory. In addition, a fuzzy sliding mode controller is developed to compensate for the fuzzy approximating errors and attenuate the interactions between subsystems. Global asymptotic stability is established in the Lyapunov sense, with the tracking errors converging to a neighborhood of zero. A simulation example is given to illustrate the performance of the proposed method.