Quasi Order Research Articles

The Set of Better Quasi Orderings is ∏

AbstractIn this paper we give a proof of the II12‐completeness of the set of countable better quasi orderings (viewed as a subset of the Cantor space). This result was conjectured by Clote in [2] and proved by the author in his Ph.d. thesis [6] (see also [7]). Here we prove it using Simpson's definition of better quasi ordering ([15]) and as little bqo theory as possible.

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order ⊴ defined onP so thata⊴b holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.

On the regularity of languages on a binary alphabet generated by copying systems

Let A be a binary alphabet and ⊢ ∗ π the derivation relation associated to the semi-Thue system π={(x, xx)∣x ∈A ∗} . We prove that ⊢ ∗ π is a well quasi order in A ∗. As a consequence of this we show that the languages which are upwards closed with respect to ⊢ ∗ π are all regular.

The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order ⊴ defined on P so that a⊴b holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is ⊴-maximal, (ii) X is a ⊴-antichain, and (iii) X is ⊴-dominating; we call such a subset a ⊴-good subset of P. A ⊴-good subset is a retract of P and any two ⊴-good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a ⊴-good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the “core” of P (a ⊴-good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.

Forecast sufficiency characteristic: Construction and application

The binary relation of sufficiency induces a quasi order among alternative forecast systems, consistent with their order in terms of economic values, yet independent of the prior distribution and the loss function. This invariance across decision problems makes the sufficiency relation an attractive tool for comparing forecasts serving multiple users. The existence of the sufficiency relation between discrete probabilistic forecasts of binary events may be verified graphically by plotting Forecast Sufficiency Characteristics (FSC). The article presents a general algorithm for constructing FSC and illustrates the concept with an application to real meteorologic forecasts.

How to build a knowledge space by querying an expert

A particular field of knowledge is conceptualized as a set of problems (or questions). A person's knowledge state in this domain is formalized as the subset of problems this person is capable of solving. When the family of all knowledge states is closed under union, it is called a knowledge space. Doignon and Falmagne (1985) established a 1-1 correspondence between knowledge spaces and a class of surmise systems, a slight variant of AND/OR graphs. Here we rather obtain a 1-1 correspondence with a well defined class of quasi orders on the collection of all subsets of problems. The resulting approach to knowledge spaces helps to build such spaces for particular domains. We describe a procedure which relies on the answers of an expert to a carefully chosen sequence of information requests.

The PT-order, minimal cutsets and menger property

If P is a poset, the associated PT-order is the quasi order ⊴ in which a ⊴ b holds if every maximal chain of P which passes through a also passes through b. P is special if whenever A is a chain in P and a=sup A or inf A, then there is b ∈ A such that b ⊴ a. It is proved that if P is chain complete and special then the set of ⊴-maximal elements is ⊴-dominating and contains a minimal cutset. As corollaries of this, we give partial answers to (i) a question of Rival and Zaguia by showing that if P is regular and special every element is in a minimal cutset and (ii) a question of Brochet and Pouzet by showing that if P is chain complete and special then it has the Menger property.

On quasi twodimensional order in XY magnets or superconductors of the second kind. Influence of multiple layering

Extending an earlier reasoning about spontaneous multiplication of interplanar vortex loops, it is suggested that multiple layering of magnetic planes, more strongly coupled within each group than from group to group, decreases strongly the two dimensional fluctuations. This should increase the three dimensional critical temperature roughly proportionately to the number of planes within each group, if other parameters remain equal. It is argued that this effect could also play an important role in the superconducting case. The situation of oxide superconductors is discussed in this light Extension d'un raisonnement concernant la multiplication spontanee de boucles de vortex interplanaires aux structures a couches multiples: un couplage plus fort entre plans magnetiques a l'interieur d'un groupe que de groupe a groupe decroit fortement les fluctuations bidimensionnelles et devrait augmenter la temperature tridimensionnelle a peu pres en proportion du nombre de plans dans chaque groupe, si les autres parametres restent constants. Discussion du role important de cet effet dans le cas de la supraconductivite de type II et des oxydes supraconducteurs

Another generalization of Higman's well quasi order result on [formula omitted

We present another generalization of Higman's result ([2]) that the subsequence embedding relationship on a finitely generated free monoid is a well quasi order. Our result is analogous to the generalization of Higman's result given in [1] in the following manner. In the quasi orders we consider, we are given a finite set S of non-null words from Σ ∗ and we demand that x 1… x k+1 is less than or equal to x 1 a 1… x k a k x k+1 for any words x 1,…,x k+1ϵΣ ∗ and any single word a 1… a k ϵ S, where a i ϵΣ, 1⩽ i⩽ k. The least quasi order obtained in this fashion is denoted ⪅ s. In contrast to the notion of subword unavoidability used in [1], S is said to be subsequence unavoidable (in Σ ∗ ) if and only if there exists a k 0 such that any word longer than k 0 has a (nonempty) subsequence of letters (not necessarily contiguous) which form a word in S. We show that ⪅ s is a well quasi order on Σ ∗ if and only if S is subsequence unavoidable in Σ ∗ . As an application of this result, we show that the iterated shuffle ([1], [7], [8]) of a finite set S with itself is a regular language if and only if S is subsequence unavoidable.

Substitutes, Complements and Ripples in Network Flows

We study the qualitative variation of minimum-cost network-flows and their associated costs with arc parameters. Demands are given at each node, flow is conserved, each arc's parameter lies in a lattice, and the flow cost is real or infinity-valued. Our main results are roughly as follows. If the flow cost is arc-additive, the problem can be decomposed into independent problems on each biconnected component of the graph. If also the flow cost is convex in the flow, the magnitude of the change in the optimal flow in arc a resulting from changing arc b's parameter diminishes the less biconnected a is to b. Arc a is less-biconnected-to b than is arc d if every simple cycle containing a and b also contains d. This relation is a quasi order with two distinct arcs being equivalent if and only if deleting them disconnects the graph. Hie Hasse diagram of the partial order of the induced equivalence classes is a tree with all arcs directed towards the class containing b. Two arcs are complements (resp., substitutes) if every simple cycle containing both orients them in the same (resp., opposite) way. For example, two arcs are conformal, i.e., complements or substitutes, if they are either incident, or lie on a common face of a planar graph, or (as Dirac [Dirac, G. A. 1952. A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27 85–92.] and Duffin [Duffin, R. J. 1965. Topology of series-parallel networks. J. Math. Anal. Appl. 10 303–318.] have shown) lie in a series-parallel graph. If also each arc cost is (lattice) subadditive, then the optimal flow in arc a is nondecreasing (resp., nonincreasing) in the parameter of each complement (resp., substitute) of a. If moreover each parameter lies in a chain, then the minimum cost is superadditive in the parameters of a set of substitutes. Suppose in addition that the arc parameters are real and the arc costs are doubly subadditive. Then the absolute difference of the optimal flows in an arc corresponding to two monotonically-step-connected parameter vectors does not exceed the sum of the absolute differences of their elements. If further the arc costs are affine between successive integers and the difference between the two parameter vectors is a unit vector, then the difference between corresponding integer optimal flows is a unit simple circulation. This fact leads to a parametric algorithm for finding optimal flows. Finally, suppose instead that the flow cost is a sum of a subadditive flow cost for arcs in a set S of complements and a flow cost for arcs not in S that is arc-additive and convex in the flows therein. Then the optimal flow in each arc in S is nondecreasing in the parameters of those arcs, and the minimum cost is subadditive therein. Moreover, the optimal flow in each arc a not in S is nondecreasing (resp., nonincreasing) in the parameters of arcs in S if a is a complement (resp., substitute) of every arc in S.

Discrete and nondlscrete quasi eon vexlike functions and single-peakedness (unimodality)

Donath and Elster's result on the equivalence of unimodal and x-convex (in the sense of Ponstein) functions in case of lower semicontinuity, together with some further results (relaxation of the semicontinuity condition etc.), is given for functions whose values are taken from a connected quasi order. The for-all-x condition can be replaced by a there-exists-anx condition. If the functions are defined on a discrete set, the equivalence only applies to a kind of unimodality where two (but not more than two) adjacent points can have the same functional value. This is the kind of unimodality that figures in Social Choise Theory in connection with the representation of single-peaked preference relations. Some 18 kinds of generalized convexity and their interrelations are discussed.

Negatively isotone optimal policies for random walk type Markov decision processes

This paper considers a random walk type Markov decision process in which the state spaceI is an integer subset of IR m , and the action spaceK is independent ofi eI. The natural order , overI, and a quasi order, ′, overK, is assumed, together with aconditional convexity assumption on the returns {r i k }, and certain other assumptions about these rewards and the transition probabilities in relationship to the orders and ′.A negatively isotone policy is one for whichi i′→δ(i)⊁′)δ(i′) (i.e.δ(i) ′δ(i)′ orδ(i′) ′δi)). It is shown that, under specified conditions, a negatively isotone optimal policy exists. Some consideration is given to computational implications in particular relationship to Howard's policy space method.

On 𝑆-closed spaces

In this paper, we initially give several new characterizations of the class of S-closed spaces, which was introduced by T. Thompson [Proc. Amer. Math. Soc. 60 (1976), 335-338]. We then employ these characterizations to produce analogues for S-closed spaces of the well-known theorem from real analysis that an upper-semicontinuous real-valued function on a closed interval assumes a maximum, and of two well-known theorems of G. Birkhoff and A. D. Wallace, which established that each upper-semicontinuous function from a compact space into a partially ordered set assumes a maximal value and that each compact space has a maximal element with respect to each upper-semicontinuous quasi order on the set. The statements in these latter analogues are then shown to characterize S-closed spaces. A “fixed set theorem” for multifunctions on S-closed spaces is also established.

Generalized Information and Data Analysis

Using the generalized information theory, we propose in this study an original approach to the problems encountered in data analysis. J. ACZEL and Z. DAROCZY have recently presented a synthesis of axiomatic approaches to classical data processing; C.F. PICARD and his research team of the CNRS on information structures have proposed various general definitions to the notion of information. The study of these works with a view to their adaptation to data analysis and to the theory of questionnaires has led us to show, with the help of an appropriate formalism, the fundamental properties of these various approaches; we have thus developed a global model: the notion of information given by a set of observers, which seems to fit a large category of data analysis problems.We propose a common formalism for the multidimensional data type Ω x 0, where Ω is a set of individuals and 0 is a set of characters (qualitative ordered or non-ordered, quantitative discrete or continuous), which are called observers in the theory of information. To eaeh observer Ω ε 0, we associate a family Fξ of parts of Ω, whose structure enables us to formalize properties of the empirical relation on Ω associated to ξ ε 0 (equivalence, membership, quasi order); we shall define on the product space ∏ξε0 (Ω,Fξ) various information measures and metrics that enable us to operate various algorithms, classical or original, without it being necessary to code the initial data table.

Calculation of a one dimensional van der waals liquid-gas transition by means of a Ginzburg-Landau expansion

The grand canonical partition function written as a Feynman path integral over the order parameter is calculated numerically within the framework of a Ginzburg-Landau theory for a one dimensional van der Waals system. The equation of state is calculated. The results show that the transition is of quasi first order.

Weakly smooth continua

We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua. In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.

A quasi order characterization of smooth continua

A quasi order characterization of smooth continua