无限集上严格保序变换半群的正则元及格林关系

设X是一个全序集,SOx是X上的严格保序变换半群.考虑X是一个全序无限集的情形.讨论了SOx中的正则元和SOx上的格林关系.证明了SOx中的元素α是正则的当且仅当α是双射.还证明了SOx的所有正则元形成SOx的一个极大子群.另外,给出了SOx上的格林关系L,R,D和H的刻划.SOx上格林关系的刻画对SOx的研究起重要作...     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

A maximal chain of principal ideals in the semigroup of binary relations on a finite set

A maximal chain of F_n+3−1 principal ideals in the semigroup of the binary relations on an n-element set X is constructed by representing a binary relation as a Boolean matrix. Here F_n stands for the n-th Fibonacci number.     

A rough set approach to intuitionistic fuzzy soft set based decision making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. Since its appearance, there has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. The intuitionistic fuzzy soft set is a combination of an intuitionistic fuzzy set and a soft set. The rough set theory is a powerful tool for dealing with uncertainty, granuality and incompleteness of knowledge in information systems. Using rough set theory, this paper proposes a novel approach to intuitionistic fuzzy soft set based decision making problems. Firstly, by employing an intuitionistic fuzzy relation and a threshold value pair, we define a new rough set model and examine some fundamental properties of this rough set model. Then the concepts of approximate precision and rough degree are given and some basic properties are discussed. Furthermore, we investigate the relationship between intuitionistic fuzzy soft sets and intuitionistic fuzzy relations and present a rough set approach to intuitionistic fuzzy soft set based decision making. Finally, an illustrative example is employed to show the validity of this rough set approach in intuitionistic fuzzy soft set based decision making problems.     

Partial order relations on family of sets and scalarizations for set optimization

In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.     

On covering an independent set in a grid with a second independent set

We show that for every independent set O in an n × m grid, n, m > 1, there is a second independent set X with the property that every member of O is adjacent to at least one member of X . The proof gives a construction for X . Equivalently, we show that for every maximal independent set in a grid, there is a second, disjoint, maximal independent set. © 1996 John Wiley & Sons, Inc.     

A set scalarization function based on the oriented distance and relations with other set scalarizations

ABSTRACT

The aim of this paper is, in the setting of normed spaces with a cone K non necessarily solid, to study new relations among set scalarization functions that are extensions of the oriented distance of Hiriart-Urruty. Moreover, we deal with a set scalarization function of sup-inf type, we investigate its relation to the cone-properness and cone-boundedness and it is related to other set scalarizations existing in the literature. In particular, with the norm induced by the Minkowski's functional, we obtain relations with a set scalarization which is an extension of the so called Gerstewitz's scalarization function.     

Relations between the convexity of a set and the differentiability of its support function

It is known that, in finite dimensions, the support function of a compact convex set with nonempty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper, we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then, we revisit some results related to the differentiability of the cost function associated to a production function.     

A hyperspatial representation of a particular set of fuzzy preference relations

In this paper we study a geometrical representation of a special class of fuzzy preference relations by establishing a one-one correspondence between the set of all these relations and a hypercube of an n(n?1)/2 dimensional space.Furthermore, we study some geometrical figures (called H-spheres) and properties of the points which represent the preference relations in the above mentioned euclidean space.     

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

On strongly pseudoconvex manifolds with one-dimensional exceptional set

Let X be a strongly pseudoconvex n-dimensional manifold with one-dimensional connected exceptional set S. Here we show that if S is reducible, then X is embeddable into C^N ×P ^{m for some N, m and in particular X is Kählerian with a possible exception for n = 3; we analyze this exceptional case but we do not know if it may occur. The case in which S is irreducible was previously analyzed by Tan [11], who proved that X is embeddable if either n ≠ 3 or S is not isomorphic to} P ¹. In the same paper, Tan gave an example of a non-embeddable and non-Kählerian strongly pseudoconvex three-dimensional manifold withP ¹ as an exceptional set.     

Extended rough set model based on known same probability dominant valued tolerance relation

Classical rough set theory is based on the conventional indiscernibility relation. It is not suitable for analyzing incomplete information. Some successful extended rough set models based on different non-equivalence relations have been proposed. The data-driven valued tolerance relation is such a non-equivalence relation. However, the calculation method of tolerance degree has some limitations. In this paper, known same probability dominant valued tolerance relation is proposed to solve this problem. On this basis, an extended rough set model based on known same probability dominant valued tolerance relation is presented. Some properties of the new model are analyzed. In order to compare the classification performance of different generalized indiscernibility relations, based on the category utility function in cluster analysis, an incomplete category utility function is proposed, which can measure the classification performance of different generalized indiscernibility relations effectively. Experimental results show that the known same probability dominant valued tolerance relation can get better classification results than other generalized indiscernibility relations.     

The stability set of voting games: Classification and genericity results

In 1980 Rubinstein introduced a new solution concept for voting games called the stability set which incorporates the idea that before entering into a possibly winning coalition with respect to some pair of alternatives, a voter will consider what might happen in the future. He showed that if the voters' preferences are given by linear orders the stability set is non-empty for a large class of voting games with finite sets of alternatives. We consider the case where indifference is allowed (preferences are then complete preorders) and show that the picture is then quite different. First, in the finite case, we obtain classification results for the non-emptiness of the stability set which are based on Nakamura's number. When preferences are continuous, we prove a general non-emptiness theorem and show that the set of profiles for which the stability set is non-empty is dense in the set of profiles.We are most grateful to the French Commissariat Général du Plan for financial support and to Georges Bordes, Prasanta K. Pattanaik, Hervé Moulin, Bezalel Peleg, Ariel Rubinstein, Norman Schofield and two anonymous referees of this journal for helpful comments. In particular, Bezalel Peleg has called our attention to the similarities of the stability set with the Gillies set and the uncovered set. A previous draft of this paper has been presented at the conference on Economic Models and Distributive Justice in Brussels, at the European Meeting of the Econometric Society in Copenhagen and at seminars in Europe and the United States. We thank the participants for their remarks.     

On the X-Y convex hull of a set of X-Y polygons

We study the class of rectilinear polygons, calledX – Y polygons, with horizontal and vertical edges, which are frequently used as building blocks for very large-scale integrated (VLSI) circuit layout and wiring. In the paper we introduce the notion of convexity within the class ofX – Y polygons and present efficient algorithms for computing theX – Y convex hulls of anX – Y polygon and of a set ofX – Y polygons under various conditions. Unlike convex hulls in the Euclidean plane, theX – Y convex hull of a set ofX – Y polygons may not exist. The condition under which theX – Y convex hull exists is given and an algorithm for testing if the given set ofX – Y polygons satisfies the condition is also presented.     

Levitin–Polyak well-posedness for set optimization problems involving set order relations

Positivity - In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set...     

Open set condition for graph directed self-similar structure

Graph directed self-similar structure generalizes the concept of self-similar set and contains some important instants of fractal sets. We characterize the open set condition (OSC), which is fundamental in the study of self-similar set, for graph directed self-similar structure in terms of the post critical set. Using this characterization, we establish the relations between OSC and other separation conditions including post-critically finite, finitely ramified condition and finite preimage property. It turns out that whether the intrinsic metric is doubling makes difference. In particular, finitely ramified condition implies OSC in case of doubling metric but does not in case of non-doubling metric.     

Union and symmetry preserving homomorphisms of semigroups of closed relations

The symbol β_X denotes the semigroup of all binary relations on a nonempty set X under composition which is defined by α_oβ={(x,y)} ∈ X×X: (x,z) ∈ β and (z,y) ∈ α for some z∈X} for all α,β ∈ β_x . In a recent paper [1, Theorem 3, p. 310], A. H. Clifford and D. D. Miller initiated a study of the endomorphisms of β_X when they completely determined those which preserve unions and take symmetric relations into symmetric relations. The purpose here is to place the theorem of Clifford and Miller in a topological setting and to discuss some of the problems which then arise naturally. The full results will appear in [9]. Partial financial support from Australian National University and the research foundation of State University of New York is gratefully acknowledged.     

The set of solutions of a proportionally modular Diophantine inequality

The set of solutions of the inequality is a numerical semigroup. We present in this paper a tool for finding the set of minimal generators of this set, and thus the set of solutions to such an inequality. This tool will also enable us to give characterizations of those numerical semigroups that are the set of integer solutions of inequalities of this form. Finally, we give a deeper study of the embedding dimension three case.     

On uniqueness of （fuzzy） relation in some generalized rough set model

In rough set theory, crisp and/or fuzzy binary relations play an important role in both constructive and axiomatic considerations of various generalized rough sets. This paper considers the uniqueness problem of the （fuzzy） relation in some generalized rough set model. Our results show that by using the axiomatic approach, the （fuzzy） relation determined by （fuzzy） approximation operators is unique in some （fuzzy） double-universe model.     

On a set of generating relations of the full transformation semigroups

A set of generating relations of the full transformation semigroup Σ_n consisting of all mappings of the set $\text{Ω = {1, …, n}}$ into itself with respect to a natural system of generators {γ_jⁱ, σ_jⁱ} (defined in Section 1) is given. The set of relations (1)～(15) given in Section 1 turns out to be a set of generating relations.