Fuzzy logic models in a category of fuzzy relations

We investigate interpretations of formulas ψ in a first order fuzzy logic in models which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations are then morphisms in a category SetR(Ω) from some Ω-set to the object . We define homomorphisms between models in a category SetR(Ω) and we prove that if is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations of a formula ψ in models . Supported by MSM6198898701, grant 201/07/0191 of GAČR and grant 1M0572.     

Constructions in the category of fuzzy T-proximity spaces

This paper is devoted to study the initial fuzzy T-proximity structures of the fuzzy T-proximity spaces defined by Hashem and Morsi in 2002, where T stands for any continuous triangular norm. In this paper, we show that all initial and final lifts in the category T-FPS of these fuzzy T-proximity spaces exist and hence the initial and final fuzzy T-proximity structures exist. We introduce a characterization for the initial fuzzy T-proximity structures, so as a special initial fuzzy T-proximity spaces, the subspaces and product spaces of these fuzzy T-proximity spaces can also be characterized. We also show that the fuzzy topology associated to the initial fuzzy T-proximity structure of a family of fuzzy T-proximity spaces, coincides with the initial fuzzy topology of the family of fuzzy topologies associated to these fuzzy T-proximity spaces.     

Fuzzy sets and cut systems in a category of sets with similarity relations

Let \Upomega\Upomega be a complete residuated lattice. Let SetR(\Upomega){\mathbf{SetR}}(\Upomega) be the category of sets with similarity relations with values in \Upomega\Upomega (called \Upomega\Upomega-sets), which is an analogy of the category of classical sets with relations as morphisms. A fuzzy set in an \Upomega\Upomega-set in the category SetR(\Upomega){\mathbf{SetR}}(\Upomega) is a morphism from \Upomega\Upomega-set to a special \Upomega\Upomega-set (\Upomega,?),(\Upomega,\leftrightarrow), where ?\leftrightarrow is the biresiduation operation in \Upomega.\Upomega. In the paper, we prove that fuzzy sets in \Upomega\Upomega-sets in the category SetR(\Upomega){\mathbf{SetR}}(\Upomega) can be expressed equivalently as special cut systems (C_a)_{a ? \Upomega}.(C_{\alpha})_{\alpha\in\Upomega}.     

Resource-bounded strong dimension versus resource-bounded category

Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resource-bounded extension: if a class X has Δ-strong dimension less than 1, then X is Δ-meager. This has the applications of explaining some of Lutz's simultaneous Δ-meager, Δ-measure 0 results and providing a new proof of a Gu's strong dimension result on infinitely-often classes.     

Product category description for web-shopping in e-commerce

In this paper, we focus on the web-shopping activity and on ways to improve the quality of the information available to consumers. We describe a tool, called a product space map, for the presentation of information about a product category which can help consumers in making purchasing decisions. Using this tool we first provide a clustering or segmentation of a product line, that is, 27-inch televisions, into price categories such as low end, moderate and high end. Once having this partitioning we then use the idea of linguistic summaries to describe the properties of each category with respect to relevant features. An example of such a summary is “Most TV's in the high price category provide extremely high resolution.” With the aid of such information it becomes much easier for consumers to understand the product line, see what they are getting for their money, and more easily and confidently locate products that are of particular value for the money. Considerable use is made of fuzzy set technology to provide the ability to describe the information in a way, using linguistic expressions, that is particularly consumer friendly. © 2001 John Wiley & Sons, Inc.     

Equiprime, 3-prime and c-prime fuzzy ideals of nearrings

In this paper, we present the notions of equiprime fuzzy ideal, 3-prime fuzzy ideal and c-prime fuzzy ideal of a nearring. We characterize these fuzzy ideals using level subsets and fuzzy points. If f: N → M is an onto nearring homomorphism, we show that the map defines a one-to-one correspondence between the set of all f-invariant (alternatively with sup property) equiprime (3-prime and c-prime, respectively) fuzzy ideals of N and the set of all equiprime (3-prime and c-prime, respectively) fuzzy ideals of M. Finally, we define fuzzy cosets determined by generalized fuzzy ideals; obtain fundamental results and isomorphism theorems.     

Oriented colorings of 2-outerplanar graphs

A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H^′ such that H has a homomorphism to H^′. In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR₆₇, which implies that their (strong) oriented chromatic number is at most 67.     

Semigroup homomorphisms and fuzzy automata

A generalized Ω-fuzzy automaton over a complete residuated lattice Ω and a monoid (M,*) and with a set S of states is introduced as a monoid homomorphism F:(M,*)→(?,°), where (?,°) is a monoid of Ω-fuzzy sets in a set S×S. An extension principle depending of proper filters Φ in Ω is introduced which then enables to introduce morphisms between generalized Ω-fuzzy automata and to introduce the category ?_Φ of these automata. It is proved that categories of classical fuzzy automata, non-deterministic automata and some other systems are equivalent to subcategories of ?_Φ for a suitable filter Φ.     

Investigations on Hotz groups for arbitrary grammars

Summary The Hotz group H(G) and the Hotz monoid M(G) of an arbitrary grammar G=(V, X, P, S) are defined by H(G)=F(VX)/P and M(G) =(VX)^*/P respectively. A language LX^* is called a language with Hotz isomorphism if there exists a grammar G with L=L(G) such that the natural homomorphism F(X)/LH(G) is an isomorphism. The main result of this paper states that homomorphic images of sentential form languages are languages with Hotz isomorphism. This is a generalization of a result of Frougny, Sakarovitch, and Valkema on context-free languages.Hotz groups are used to obtain lower bounds for the number of productions which are needed to generate a language. Further it is shown that there are languages with Hotz isomorphism without being a homomorphic image of a sentential form language, and there are context-sensitive languages without Hotz isomorphism. The theory of Hotz monoids is used to get some results on languages generated by grammars with a symmetric set of rules.     

Introducing the mathematical category of artificial perceptions

Perception is the recognition of elements and events in the environment, usually through integration of sensory impressions. It is considered here as a broad, high-level, concept (different from the sense in which computer vision/audio research takes the concept of perception). We propose and develop premises for a formal approach to a fundamental phenomenon in AI: the diversity of artificial perceptions. A mathematical substratum is proposed as a basis for a rigorous theory of artificial perceptions. A basic mathematical category is defined. Its objects are perceptions, consisting of world elements, connotations, and a three-valued (true, false, undefined) predicative correspondence between them. Morphisms describe paths between perceptions. This structure serves as a basis for a mathematical theory. This theory provides a way of extending and systematizing certain intuitive pre-theoretical conceptions about perception, about improving and/or completing an agent's perceptual grasp, about transition between various perceptions, etc. Some example applications of the theory are analyzed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fuzzy groups: Some group-theoretic analogs

We prove a number of results about fuzzy groups involving the concepts of fuzzy cosets and fuzzy normal subgroups which are analogs of important results from group theory. Also, we introduce analogs of some group-theoretic concepts such as characteristic subgroup, normalizer, and Abelian groups. We prove that if μ is a fuzzy subgroup of a group G such that the fuzzy index of μ is the smallest prime dividing the order of G, then μ is a fuzzy normal subgroup. Also, we show that there is a one-to-one correspondence between the fuzzy (right) cosets of a fuzzy subgroup μ of a group G and the (right) cosets of a certain subgroup H of G.     

The completeness and applications of the formal system ℒ<Superscript>*</Superscript>

Since the formal deductive system ℒ^* was built up in 1997, in has played important roles in the theoretical and applied research of fuzzy logic and fuzzy reasoning. But, up to now, the completeness problem of the system ℒ^* is still an open problem. In this paper, the properties and structure ofR ₀ algebras are further studied, and it is shown that every tautology on theR ₀ interval [0,1] is also a tautology on anyR ₀ algebra. Furthermore, based on the particular structure of ℒ^*-Lindenbaum algebra, the completeness and strong completeness of the system ℒ^* are proved. Some applications of the system ℒ^* in fuzzy reasoning are also discussed, and the obtained results and examples show that the system ℒ^* is suprior to some other important fuzzy logic systems.     

格的模糊同态

研究模糊同态,自然离不开模糊映射,群与环的模糊同态分别利用了不同的模糊映射进行讨论。利用经典集间的模糊映射,引入了格的模糊同态和模糊弱同态概念,给出了模糊同态与模糊弱同态下模糊子格（模糊理想）的对应关系,并由模糊同态得到了格的理想,建立了格的同构映射,基于模糊同态所确定的映射等价刻画了模糊子格的弱（满）同态。     

群上的模糊软同余

为了进一步研究模糊软集理论,通过将软集与模糊同余结合,给出了模糊软同余的定义,并在一定程度上推广了模糊软集。研究了群上的模糊软同余的相关性质。建立了群上的模糊软同余与正规模糊软群之间的联系,给出了软模糊同态定理。     

Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems

This paper presents a novel quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems via H^∞ approach. In the proposed algorithm, a self-constructing neural fuzzy network (SCNFN) is developed with both structure and parameter learning phases, so that the number of fuzzy rules and network parameters can be adaptively determined. Based on the SCNFN, an uncertainty observer is first introduced to watch compound system uncertainties. Subsequently, an optimal NFN-based controller is designed to overcome the effects of unstructured uncertainty and approximation error by integrating the NFN identifier, linear optimal control and H^∞ approach as a whole. The adaptive tuning laws of network parameters are derived in the sense of quadratic stability technique and Lyapunov synthesis approach to ensure the network convergence and H^∞ synchronization performance. The merits of the proposed control scheme are not only that the conservative estimation of NFN approximation error bound is avoided but also that a suitable-sized neural structure is found to sufficiently approximate the system uncertainties. Simulation results are provided to verify the effectiveness and robustness of the proposed control method.     

Multiple attribute group decision-making based on order-α divergence and entropy measures under q-rung orthopair fuzzy environment

The q-rung orthopair fuzzy set (^qROPFS), proposed by Yager, is a more effective and proficient tool to represent uncertain or vague information in real-life situations. Divergence and entropy are two important measures, which have been extensively studied in different information environments, including fuzzy, intuitionistic fuzzy, interval-valued fuzzy, and Pythagorean fuzzy. In the present communication, we study the divergence and entropy measures under the q-rung orthopair fuzzy environment. First, the work defines two new order-α divergence measures for ^qROPFSs to quantify the information of discrimination between two ^qROPFSs. We also examine several mathematical properties associated with order-α ^qROPF divergence measures in detail. Second, the paper introduces two new parametric entropy functions called “order-α ^qROPF entropy measures” to measure the degree of fuzziness associated with a ^qROPFS. We show that the proposed order-α divergence and entropy measures include several existing divergence and entropy measures as their particular cases. Further, the paper develops a new decision-making approach to solve multiple attribute group decision-making problems under the ^qROPF environment where the information about the attribute weights is completely unknown or partially known. Finally, an example of selecting the best enterprise resource planning system is provided to illustrate the decision-making steps and effectiveness of the proposed approach.     

Rare category exploration

Rare category discovery aims at identifying unlabeled data examples of rare categories in a given data set. The existing approaches to rare category discovery often need a certain number of labeled data examples as the training set, which are usually difficult and expensive to acquire in practice. To save the cost however, if these methods only use a small training set, their accuracy may not be satisfactory for real applications. In this paper, for the first time, we propose the concept of rare category exploration, aiming to discover all data examples of a rare category from a seed (which is a labeled data example of this rare category) instead of from a training set. To this end, we present an approach known as the FRANK algorithm which transforms rare category exploration to local community detection from a seed in a kNN (k-nearest neighbors) graph with an automatically selected k value. Extensive experimental results on real data sets verify the effectiveness and efficiency of our FRANK algorithm.     

Fuzzy relations and fuzzy groups

Given a group S, we consider fuzzy relations on S, that is, maps from S × S into [0,1]. Of particular interest is to investigate conditions under which the fuzzy relation becomes a fuzzy subgroup on S × S. We prove that if σ is a fuzzy subset of S and μ_σ is the strongest fuzzy relation on S that is a fuzzy relation on σ, then μ_σ is a fuzzy subgroup if and only if σ is a fuzzy subgroup. A number of other results are obtained about the interrelationships between fuzzy relations on S (including the weakest fuzzy relation) and fuzzy subgroups on S × S.     

Constructing normalisers in finite soluble groups

This paper describes algorithms for constructing a Hall π-subgroup H of a finite soluble group G and the normaliser N_G(H). If G has composition length n, then H and N_G(H) can be constructed using O(n⁴ log |G|) and O(n⁵ log |G|) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative system normalisers. Computer implementations of these algorithms can compute a Sylow 3-subgroup of a group with n = 84, and its normaliser in 47 seconds and 30 seconds, respectively. Constructing normalisers of arbitrary subgroups of a finite soluble group can be complicated. This is shown by an example where constructing a normaliser is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms.     

Fuzzy hyperlattices

In this paper, we introduce the notion of fuzzy hyperlattices and establish connections between fuzzy hyperlattices and hyperlattices. Then we discuss the relations between the weak homomorphism of fuzzy hyperlattices and the weak homomorphism of corresponding hyperlattices. Moreover, we define and analyze the hypercongruence on hyperlattices and the fuzzy (strong) hypercongruence on fuzzy hyperlattices. And we study the interrelation between the (fuzzy) hypercongruence on (fuzzy) hyperlattices and the homomorphism of (fuzzy) hyperlattices. Also, a relation between the fuzzy hypercongruence on fuzzy hyperlattices and the hypercongruence on corresponding hyperlattices is obtained. Finally, we apply the hypercongruence and the fuzzy strong hypercongruence to construct quotient structures of hyperlattices and fuzzy hyperlattices, respectively.