Coloring linear orders with Rado's partial order

Let ?_R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order (a standard example of a wqo which is not a bqo). We show that ?_R has fairly high complexity with respect to Borel reducibility (e.g. if P is a Borel preorder, then P ≤_B ?_R), although its exact classification remains open. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decomposition of partial orders

A decomposition theory for partial orders which arises from the split decomposition of submodular functions is introduced. As a consequence of this theory, any partial order has a unique decomposition consisting of indecomposable partial orders and certain highly decomposable partial orders. The highly decomposable partial orders are completely characterized. As a special case of partial orders, we consider lattices and distributive lattices. It occurs, that the highly decomposable distributive lattices are precisely the Boolean lattices.     

Nests with the partial factorization property

It is proved that a nest on a separable complex Hilbert space has the left (resp. right) partial factorization property, which means that for every invertible operator from onto a Hilbert space there exists an isometry (resp. a coisometry) from into such that both and are in the associated nest algebra if and only if it is atomic (resp. countable).

     

Global optimization in metric spaces with partial orders

The main objective of this article is to resolve an optimization problem in the setting of a metric space that is endowed with a partial order. In fact, given non-empty subsets A and B of a metric space that is equipped with a partial order, and a non-self mapping S: A?→?B, this article explores the existence of an optimal approximate solution, known as a best proximity point of the mapping S, to the equation Sx?=?x, where S is a proximally increasing, ordered proximal contraction. This article exhibits an algorithm for determining such an optimal approximate solution. Moreover, the result elicited in this article subsumes a fixed point theorem, due to Nieto and Rodriguez-Lopez, in the setting of a metric space with a partial order.     

Asymptotic enumeration ofN-free partial orders

LetN(n) andN ^* (n) denote, respectively, the number of unlabeled and labeledN-free posets withn elements. It is proved thatN(n)=2 ^{n logn+o(n logn)} andN ^*(n)=2^{2n logn+o(n logn)}. This is obtained by considering the class ofN-free interval posets which can be easily counted.     

Generalized o-minimality for partial orders

We consider partially ordered models. We introduce the notions of a weakly (quasi-)p.o.-minimal model and a weakly (quasi-)p.o.-minimal theory. We prove that weakly quasi-p.o.-minimal theories of finite width lack the independence property, weakly p.o.-minimal directed groups are Abelian and divisible, weakly quasi-p.o.-minimal directed groups with unique roots are Abelian, and the direct product of a finite family of weakly p.o.-minimal models is a weakly p.o.-minimal model. We obtain results on existence of small extensions of models of weakly quasi-p.o.-minimal atomic theories. In particular, for such a theory of finite length, we find an upper estimate of the Hanf number for omitting a family of pure types. We also find an upper bound for the cardinalities of weakly quasi-p.o.-minimal absolutely homogeneous models of moderate width.     

Some inequalities for partial orders

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most maximal chains, if some antichain has cardinality v, and if there are ₁ chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ₁v (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).This paper was written while the authors were visitors at the Laboratoire d'algèbre ordinale, Département de Mathématiques, Université Claude Bernard, Lyon 1, France.Research supported by NSERC grant # A5198.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

A Ramsey theorem for partial orders with linear extensions

We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.     

On the Ferrers property of valued interval orders

We study the relationship between the Ferrers property and the notion of interval order in the context of valued relations. Given a crisp preference structure without incomparability, the strict preference relation satisfies the Ferrers property if and only if the associated weak preference relation does. These conditions characterize a total interval order. For valued relations the Ferrers property can be written in two different and non-equivalent ways. In this work, we compare these properties by finding the kind of completeness they imply. Moreover, we study whether they still characterize a valued total interval orders.     

Directed partial orders on real quaternions

It is shown that a division ring of real quaternions can be made into a partially ordered ring with a directed partial order.     

Directed partial orders over non-archimedean o-fields

Positivity - Let F be a non-archimedean o-field, $$C=F(i)$$ the imaginary quadratic extension field of F with $$i^2=-1$$ . In this paper, all directed partial orders on C are classified via the new...     

Orthogonal structures on finite partial orders

By introducing an inner product on the linear space of real valued functions on a finite partial order, broad generalizations of Sperner's lemma are obtained. In the special case of a power set, the subspace spanned by the maximal chains is characterized. Finally, applications of the general theorem to the theory of numbers and to the theory of probability are given without proof.     

Product innovation with partial capacity rollover

Central European Journal of Operations Research - This paper analyzes how the transferability of production capacities from an established to a new product influences the incentives of a firm to...     

A comparative analysis of methods for constructing weak orders from partial orders

This paper compares three methods (sequel, cardinal, maximal) for constructing a weak order from a partial order on a finite set. The constructed weak orders include the partial order. To evaluate the methods, several different selection disciplines were used to stochastically generate partial orders from a fixed linear order. The error of a weak order which includes a generated partial order is a function of the number of ordered pairs added to the partial order to get the weak order which are the reverse of ordered pairs in the fixed linear order. In all cases, the sequel and cardinal mean errors were much lower than the maximal mean error. In most but not all cases, the cardinal mean error was lower than the sequel mean error.