Szpilrajn Theorem Research Articles

Splitting metrics by [formula omitted]-quasi-metrics

It is well known that by Szpilrajn's Theorem each partial order on a set can be extended to a linear order. In this paper we study variants of that theorem that hold for T0-quasi-metrics instead of partial orders.More specifically, given a metric space (X,m), we call a T0-quasi-metric d defined on the set X m-splitting provided that its symmetrization ds=d∨d−1 is equal to m.For a given metric space (X,m) we obtain results about T0-quasi-metrics on X that are minimal among the collection of all m-splitting T0-quasi-metrics on X.We are also interested in the maximal possible specialization orders of m-splitting T0-quasi-metrics.

Conditional extensions of fuzzy preorders

The problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations and their applications. Several variations of the acclaimed Szpilrajn's theorem have been provided, inclusive of the case when some order conditions between elements are imposed on the extension. We extend the analysis of that topic by Alcantud to the fuzzy case. By appealing to generators to decompose (fuzzy) preference relations into strict preference and indifference relations, we give general extension results for the corresponding concept of compatible extension of a fuzzy reflexive relation. Then we investigate the conditions under which compatible order extensions exist such that certain elements are connected by the asymmetric part, resp., and certain other elements by the symmetric part, to respective elements with degree 1.

SZPILRAJN, ARROW AND SUZUMURA: CONCISE PROOFS OF EXTENSION THEOREMS AND AN EXTENSION

ABSTRACTThis paper extends the classical extension theorem established by Edward Szpilrajn (Fundamenta Mathematicae, 16, pp. 386–389, 1930). Szpilrajn's theorem states that every quasi‐ordering has an ordering extension. Because of its usefulness in various themes of economics, it has been applied by many researchers. Important generalizations have been presented by two authors, Kenneth Arrow and Kotaro Suzumura, among others. First, we provide concise proofs of four extension theorems by Szpilrajn, Arrow and Suzumura. We then show an extension of their extension theorems.

Fuzzy Szpilrajn Theorem through Indicators

In this paper there are studied some numerical indicators which measure the degree to which a fuzzy relation verifies some properties (reflexivity, transitivity, etc. ).The main result is a fuzzy generalization of the Szpilrajn theorem in terms of such numerical indicators and is applied to any fuzzy relation.

On a Possible Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

The Szpilrajn theorem and its strengthening by Dushnik and Miller belong to the most quoted theorems in many fields of pure and applied mathematics as, for instance, order theory, mathematical logic, computer sciences, mathematical social sciences, mathematical economics, computability theory and fuzzy mathematics. The Szpilrajn theorem states that every partial order can be refined or extended to a total (linear) order. The theorem by Dushnik and Miller states, moreover, that every partial order is the intersection of its total (linear) refinements or extensions. Since in mathematical social sciences or, more general, in any theory that combines the concepts of topology and order one is mainly interested in continuous total orders or preorders in this paper some aspects of a possible continuous analogue of the Szpilrajn theorem and its strengthening by Dushnik and Miller will be discussed. In particular, necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Dushnik-Miller theorem will be presented. In addition, it will be proved that a continuous analogue of the Szpilrajn theorem does not hold in general. Further, necessary and in some cases necessary and sufficient conditions for a topological space to satisfy a possible continuous analogue of the Szpilrajn theorem will be presented.

On a Strong Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

On a Strong Continuous Analogue of the Szpilrajn Theorem and its Strengthening by Dushnik and Miller

Compatible extensions of fuzzy relations

In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999). Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results. In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan’s theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive. Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter’s theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.

On the continuous analogue of the Szpilrajn Theorem I

One of the best known theorems in order theory, mathematical logic, computer sciences and mathematical social sciences is the Szpilrajn Theorem which states that every partial order can be refined to a linear order. Since in mathematical social sciences one frequently is interested in continuous linear orders or preorders, in this paper the continuous analogue of the Szpilrajn Theorem will be discussed.

The Connection Between Demand and Utility

Relations by which demand and utility are connected have variants and alternative expressions; and parts generally independent become interdependent under assumptions about the utility. This investigation serves to elaborate and systemize principal phases in the development of such relations. A review about binary relations and orders, which includes Szpilrajn's theorem on order refinement, is followed by an account of efficiency concepts relevant to the demand-utility relationship. Then follows a study of properties of the utility-cost function, basic to McKenzie's approach to necessity of Slutsky conditions, to be consolidated and joined with a treatment of the more complex sufficiency side. Introduction of the normal demand function, and normalization of the standard demand function, is preliminary to elaborations on the revealed preference principle of Samuelson, extended by Houthakker. Remaining topics concern general demand correspondences, direct and indirect utility, duality, and the canonical order.

Fundamentals of modelling concurrency using discrete relational structures

We consider relational structures \((X,R_1,R_2)\) such that \(X\) is a set and \(R_1,R_2\) are two binary relations on \(X\). For a number of different classes of structures we show that any structure can be represented as the intersection of its maximal extensions. Such a property – called extension completeness – can be seen as a generalisation of Szpilrajn's theorem which states that each partial order is the intersection of its total order extensions. When \(R_1\) can be interpreted as causality and \(R_2\) as ‘weak’ causality we obtain a model of concurrent histories generalising that based on causal partial orders.

Partial ordering in L-underdeterminate sets

The purpose of this paper is to improve results on fuzzy partial orderings obtained by Zadeh in [9]. We overcome the difficulties connected with the axioms of antisymmetry and linearity. Moreover, if the underlying lattice L is a complete residuated lattice, we establish a Szpilrajn theorem, i.e., any ( L-fuzzy) partial ordering has a linear extension. In opposition to Zadeh's, our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless. Therefore we first introduce the category LUS (cf. [2]), which can be considered as a mathematical model of fuzzy equality, and subsequently we specify the axioms of ( L-fuzzy) partial orderings with respect to the frame given by LUS. The axioms we use clearly display the usefulness of having a Zadeh-like complementation and, as a consequence, the usefulness of a positivistic (and nonintuitionistic) frame of study. An example concerning L°( R n ) which we give clearly shows that the LUS version of the Szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem, but provides us with additional information.

Szpilrajn's theorem on fuzzy orderings

Fuzzy relation matrices are considered and the well-known Szpilrajn theorem on orderings is generalized to fuzzy orderings. Zadeh has already shown a generalized theorem, but another generalization is given in matrix form. Some interesting operations on fuzzy matrices are introduced, and their properties are mentioned briefly. The theorem is represented in terms of the matrix operations. The operations have many properties and are useful in the various applications of fuzzy matrices or fuzzy relations.

Similarity relations and fuzzy orderings

The notion of “similarity” as defined in this paper is essentially a generalization of the notion of equivalence. In the same vein, a fuzzy ordering is a generalization of the concept of ordering. For example, the relation x ≫ y ( x is much larger than y) is a fuzzy linear ordering in the set of real numbers. More concretely, a similarity relation, S, is a fuzzy relation which is reflexive, symmetric, and transitive. Thus, let x, y be elements of a set X and μ s(x,y) denote the grade of membership of the ordered pair ( x,y) in S. Then S is a similarity relation in X if and only if, for all x, y, z in X, μ s(x,x) = 1 (reflexivity), μ s(x,y) = μ s(y,x) (symmetry), and μ s(x,z) ⩾ ∨ (μ s(x,y) Å μ s(y,z)) (transitivity), where ∀ and Å denote max and min, respectively. y A fuzzy ordering is a fuzzy relation which is transitive. In particular, a fuzzy partial ordering, P, is a fuzzy ordering which is reflexive and antisymmetric, that is, ( μ P(x,y) > 0 and x ≠ y) ⇒ μ P(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x ≠ y ⇒ μ s(x,y) > 0 or μ s(y,x) > 0. A fuzzy preordering is a fuzzy ordering which is reflexive. A fuzzy weak ordering is a fuzzy preordering in which x ≠ y ⇒ μ s(x,y) > 0 or μ s(y,x) > 0. Various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, an extended version of Szpilrajn's theorem is proved.