On consistency measures of linguistic preference relations

Inspired by the concept of deviation measure between two linguistic preference relations, this paper further defines the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Based on this, we present a consistency index of linguistic preference relations and develop a consistency measure method for linguistic preference relations. This method is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons using the linguistic label set. Using this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations, and also investigate the consistency properties of collective linguistic preference relations. These results are of vital importance for group decision making with linguistic preference relations.     

Classes of finite relations as initial abstract data types I

The aim of this paper is to give axiomatizations for sixteen types of finite relations. These classes of relations are obtained as intersections of the following basic classes of relations: total relations, surjective relations, partial functions, and injective relations.

A normal form for all relations is given and each of the sixteen types of relations is (syntactically) characterized by certain additional conditions on this normal form.

For each of the sixteen types T, a set of identities E_T is singled out. The class of relations of type T forms an initial algebra in the category of all algebras which satisfy E_T. In the first part of this paper, for each type T the involved algebras are symmetric strict monoidal categories (in the sense of MacLane), enriched with certain specific constants.     

Local methods for Rosenberg relations

In this paper, we develop local methods for studying the structure of the weak Krasner algebras generated by Rosenberg relations. In particular, this gives a complete understanding of the distributive lattices of m-ary relations in these algebras. Such knowledge is crucial for the enumeration of all relations whose endomorphism monoid is a supermonoid of the endomorphism monoid of a Rosenberg relation.     

Logics of Complementarity in Information Systems

Each information system (or data table) leads to a hierarchy of binary relations on the object set in a natural way; these relational systems can serve as frames for the semantics of modal logics. While relations of indiscernibility and their logics have been frequently studied, the situation in the case of relations which distinguish objects is much less clear. In this paper, we present complete logical systems for relations of complementarity derived from information systems.     

A system of recurrence relations for rational approximations of the Euler constant

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers.     

Logics for stable and unstable mereological relations

In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices.     

Structure relations for monic orthogonal polynomials in two discrete variables

In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector–matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.     

A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.     

From Petri Nets to Automata with Concurrency

Automata with concurrency relations are labelled transition systems with a collection of state-dependent binary independence relations for the actions. We show how to associate with each Petri net (place/transition net) such an automaton having the same dynamic behaviour. We characterize the automata arising in this way, and with suitable notions of morphisms for Petri nets and for automata with concurrency relations we extend this correspondence to a coreflection between the associated categories. As a consequence, we derive that these categories have products and conditional coproducts.     

Stochastic Relations and the Problem of Prior in the Principle of Maximum Entropy

In this paper we discuss the problem of prior for the maximum entropy principle. We show that stochastic relations can be used to constrain priors and in some case uniquely determine them. The principle of maximum entropy turns stochastic relations into (over)determined systems of partial difference equations for the partition function. All statistical consequences of the stochastic relations are determined by the space of solutions of the system.     

二元关系的完全计数

根据二元关系与其矩阵的关系,讨论了满足特殊性质的二元关系的完全计数.特别是以整数分拆的办法得到二元等价关系的完全计数.     

A note on the moments of order statistics from doubly truncated exponential distribution

Summary In a recent paper [2], the author has obtained some recurrence relations between the moments of order statitics from the exponential and right truncated exponential distributions. In this paper, similar relations are derived for a doubly truncated exponential distribution. It is shown that one can obtain all the moments by using these recurrence relations.     

On Isomorphisms of Relations Embedded into Each Other

In this paper, we consider analogs of the Cantor--Bernstein theorem for sets with binary relations. In Sec. 1, we prove an analog of this theorem for arbitrary binary relations; in Sec. 2, we consider an application; in Sec. 3, we study a class of relations with the Cantor-- Bernstein property and a class of exact relations, and prove that these classes are closed under certain operations.     

Comparison of Existence Results for Efficient Points

Existence results of maximal points with respect to general binary relations were stated by Hazen and Morin (Ref. 1) and by Gajek and Zagrodny (Ref. 2). In this paper, we point out that the natural framework for this problem is that of transitive and reflexive relations (preorders). The aim of this paper is to discuss existence results for maximal points with respect to general transitive relations in such a way that, when considering them for preorders defined by convex cones, we are able to recover most known existence results for efficient points; the quasi-totality of them, with their (short) proofs, is presented, too.     

Some new proofs of modular relations for the Göllnitz-Gordon functions

In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter’s formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.     

Dissipative extension theory for linear relations

This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning particular realizations of this general setting.     

New modular relations involving cubes of the Göllnitz–Gordon functions

Chen and Huang established some elegant modular relations for the Göllnitz–Gordon functions analogous to Ramanujan’s list of forty identities for the Rogers–Ramanujan functions. In this paper, we derive some new modular relations involving cubes of the Göllnitz–Gordon functions. Furthermore, we also provide new proofs of some modular relations for the Göllnitz–Gordon functions due to Gugg.     

Polylogarithms and multiple zeta values from free Rota-Baxter algebras

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series. We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.     

An approach to group choice with fuzzy preference relations

This paper presents an approach to group choice with fuzzy preference relations. The experts' judgements are presented as fuzzy preorder relations. As group choice we assume the nearest relation to these relations, in the sense of the Hamming distance. In this paper an algorithm for finding a preorder relation as an expression of group choice is given and a way is pointed out to find the best alternative on the basis of this group choice. Furthermore, the connection between the impossibility theorem of Arrow and this group choice is considered.     

The structure of relations in personnel management

This paper addresses the problem of defining and analyzing relations between finite sets which are involved in personnel management.Personnel management criteria are imprecise due to the complex nature of the requirements and the difficulties to deal with personnel characteristics. Fuzzy set theory seems to be an efficient tool for considering these imprecisions. According to this idea the relations involved in personnel management can be seen as fuzzy relations.R.H. Atkin has formulated the abstract simplicial complex, achieving a structure which permits a deep knowledge of non fuzzy relations. In a sense this structure can be interpreted as a geometrical multidimensional one. Our purpose is to set up a multidimensional structure associated with the fuzzy relations which appear in personnel management.By means of an analysis of the above structure it is possible to study problems concerning the recruitment selection and promotion of personnel.