### فهرست مطالب

• Volume:15 Issue:2, 2018
• تاریخ انتشار: 1397/02/05
• تعداد عناوین: 10
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• D. R. Jardon, M. Sanchis Pages 1-21
We study the space of all continuous fuzzy-valued functions from a space $X$ into the space of fuzzy numbers $(\mathbb{E}\sp{1},d\sb{\infty})$ endowed with the pointwise convergence topology. Our results generalize the classical ones for continuous real-valued functions. The field of applications of this approach seems to be large, since the classical case allows many known devices to be fitted to general topology, functional analysis, coding theory, Boolean rings, etc.
Keywords: Fuzzy-number, Fuzzy analysis, Function space, Pointwise convergence, Dual map, Evaluation map, Frechet space, Grothendieck's theorem, Cardinal function
• Chong Shen, Fu-Gui Shi Pages 23-40
The concepts of $L$-convex systems and Scott-hull spaces are proposed on frame-valued setting. Also, we establish the categorical isomorphism between $L$-convex systems and Scott-hull spaces. Moreover, it is proved that the category of $L$-convex structures is bireflective in the category of $L$-convex systems. Furthermore, the quotient systems of $L$-convex systems are studied.
Keywords: $L$-convex system, Scott-hull space, Induced $L$-convex structure, Quotient system
• Shao-Jun Yang, Fu-Gui Shi Pages 41-52
In this paper, as an application of fuzzy matroids, the fuzzifying greedy algorithm is proposed and an achievable example is given. Basis axioms and circuit axioms of fuzzifying matroids, which are the semantic extension for the basis axioms and circuit axioms of crisp matroids respectively, are presented. It is proved that a fuzzifying matroid is equivalent to a mapping which satisfies the basis axioms or circuit axioms.
Keywords: Fuzzifying matroid, Fuzzifying base-map, Fuzzifying basis axiom, Fuzzifying circuit-map, Fuzzifying circuit axiom
• Radek Slesinger Pages 53-73
Based on the notion of $Q$-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of $Q$-sup-algebras -- $Q$-sup-lattices endowed with a collection of finitary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in \cite{zhang-laan}, we characterize their subalgebras and quotients, and following \cite{solovyov-qa}, we show that the category of $Q$-sup-algebras is isomorphic to a certain subcategory of a category of $Q$-modules.
Keywords: $Q$-order, $Q$-sup-lattice, $Q$-ordered algebra, $Q$-sup-algebra, Quotient, Subalgebra
• Zhen-Yu Xiu, Bin Pang Pages 75-87
Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $\mathscr{B}$ (resp. $\varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with $\mathscr{B}$ (resp. $\varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbases can be used to characterize CP mappings and CC mappings between $M$-fuzzifying convex spaces.
Keywords: $M$-fuzzifying convex structure, Base axiom, Subbase axiom, CP mapping, CC mapping
• Hui Li, Bo Zhang, Jin Peng Pages 89-108
Uncertain graphs are employed to describe graph models with indeterministic information that produced by human beings. This paper aims to study the maximum matching problem in uncertain graphs. The number of edges of a maximum matching in a graph is called matching number of the graph. Due to the existence of uncertain edges, the matching number of an uncertain graph is essentially an uncertain variable. Different from that in a deterministic graph, it is more meaningful to investigate the uncertain measure that an uncertain graph is $k$-edge matching (i.e., the matching number is greater than or equal to $k$). We first study the properties of the matching number of an uncertain graph, and then give a fundamental formula for calculating the uncertain measure. We further prove that the fundamental formula can be transformed into a simplified form. What is more, a polynomial time algorithm to numerically calculate the uncertain measure is derived from the simplified form. Finally, some numerical examples are illustrated to show the application and efficiency of the algorithm.
Keywords: Uncertainty theory, Uncertain measure, Maximum matching, Matching number, Uncertain graph
• Hassan Dana Mazraeh, Ali Abbasi Molai Pages 109-133
This paper studies the nonlinear optimization problems subject to bipolar max-min fuzzy relation equation constraints. The feasible solution set of the problems is non-convex, in a general case. Therefore, conventional nonlinear optimization methods cannot be ideal for resolution of such problems. Hence, a Genetic Algorithm (GA) is proposed to find their optimal solution. This algorithm uses the structure of the feasible domain of the problems and lower and upper bound of the feasible solution set to choose the initial population. The GA employs two different crossover operations: 1- N-points crossover and 2- Arithmetic crossover. We run the GA with two crossover operations for some test problems and compare their results and performance to each other. Also, their results are compared with the results of other author's works.
Keywords: Bipolar fuzzy relation equations, Max-min composition, Nonlinear optimization, Genetic algorithm
• Cuilian You, Na Xiang Pages 133-142
Uncertainty theory is a mathematical methodology for dealing with non-determinate phenomena in nature. As we all know, uncertain process and uncertain integral are important contents of uncertainty theory, so it is necessary to have deep research. This paper presents the definition and discusses some properties of strong comonotonic uncertain process. Besides, some useful formulas of uncertain integral such as nonnegativity, monotonicity, intermediate results are studied.
Keywords: Uncertain variable, Uncertain process, Uncertain integral, Monotonicity
• J. Mockor Pages 143-163
Powerset structures of extensional fuzzy sets in sets with similarity relations are investigated. It is proved that extensional fuzzy sets have powerset structures which are powerset theories in the category of sets with similarity relations, and some of these powerset theories are defined also by algebraic theories (monads). Between Zadeh's fuzzy powerset theory and the classical powerset theory there exists a strong relation, which can be represented as a homomorphism. Analogical results are also proved for new powerset theories of extensional fuzzy sets.
Keywords: Extensional fuzzy sets, Powerset operators, Monads in categories
• S. P. Tiwari, I. Perfilieva, A. P. Singh Pages 165-182
The aim of the present work is to study the $F$-transform over a generalized residuated lattice. We discuss the properties that are common with the $F$-transform over a residuated lattice. We show that the $F^{\uparrow}$-transform can be used in establishing a fuzzy (pre)order on the set of fuzzy sets.
Keywords: Generalized residuated lattice, Fuzzy partition, Direct $F$-transform, Inverse $F$-transform