scholarly journals Ramsey Properties of Countably Infinite Partial Orderings

10.37236/3151 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Marcia J. Groszek
Keyword(s):  
Natural Number ◽  
Large Class ◽  
Bipartite Graphs ◽  
Partial Ordering ◽  
Partial Orderings ◽  
Countably Infinite

A partial ordering $\mathbb P$ is chain-Ramsey if, for every natural number $n$ and every coloring of the $n$-element chains from $\mathbb P$ in finitely many colors, there is a monochromatic subordering $\mathbb Q$ isomorphic to $\mathbb P$.  Chain-Ramsey partial orderings stratify naturally into levels.  We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs.  A similar analysis applies to a large class of countably infinite partial orderings with infinite levels.

Natural Partial Orders

1968 ◽  
Vol 20 ◽  
pp. 535-554 ◽  
Author(s):  
R. A. Dean ◽  
Gordon Keller
Keyword(s):  
Partial Orders ◽  
Partial Ordering ◽  
Finite Set ◽  
Partial Orderings

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).

Monotone but not positive subsets of the Cantor space

1987 ◽  
Vol 52 (3) ◽  
pp. 817-818 ◽  
Author(s):  
Randall Dougherty
Keyword(s):  
Partial Ordering ◽  
Background Information ◽  
Countable Union ◽  
Abstract Form ◽  
Cantor Space ◽  
Countable Intersection ◽  
Power Set ◽  
Countably Infinite ◽  
Infinite Set

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

2002 ◽  
Vol 16 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Dilip Roy
Keyword(s):  
Present Article ◽  
Failure Rate ◽  
Partial Ordering ◽  
Increasing Failure Rate ◽  
Multivariate Generalization ◽  
Partial Orderings ◽  
Life Distributions ◽  
New Better Than Used ◽  
Better Than

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

Partial Orderings of Distributions Based on Right-Spread Functions

1998 ◽  
Vol 35 (1) ◽  
pp. 221-228 ◽  
Author(s):  
J. M. Fernandez-Ponce ◽  
S. C. Kochar ◽  
J. Muñoz-Perez
Keyword(s):  
Probability Distributions ◽  
Partial Ordering ◽  
Dispersion Measure ◽  
Dispersive Ordering ◽  
Partial Orderings

In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive ordering and it retains most of its interesting properties.

A new ordering for stochastic majorization: theory and applications

1992 ◽  
Vol 24 (03) ◽  
pp. 604-634 ◽  
Author(s):  
Cheng-Shang Chang
Keyword(s):  
Sample Path ◽  
Stochastic Ordering ◽  
Partial Ordering ◽  
Scheduling Problems ◽  
Unified Approach ◽  
Finite Buffers ◽  
Routing And Scheduling ◽  
Partial Orderings ◽  
Stochastic Load ◽  
Stochastic Majorization

In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.

scholarly journals Representation functions of additive bases for abelian semigroups

2004 ◽  
Vol 2004 (30) ◽  
pp. 1589-1597 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Large Class ◽  
Abelian Semigroup ◽  
Representation Function ◽  
Asymptotic Basis ◽  
Countably Infinite ◽  
The Given

A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most finitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is given function from the semigroup into the set of nonnegative integers together with infinity such that this function has only finitely many zeros. It is proved that for a large class of countably infinite abelian semigroups, there exists a basis whose representation function is exactly equal to the given function for every element in the semigroup.

Some results on the relative ageing of two life distributions

1994 ◽  
Vol 31 (4) ◽  
pp. 991-1003 ◽  
Author(s):  
Debasis Sengupta ◽  
Jayant V. Deshpande
Keyword(s):  
Hazard Ratio ◽  
Partial Ordering ◽  
Closure Properties ◽  
Partial Orderings ◽  
Life Distributions ◽  
Crossing Hazards ◽  
Moment Bounds

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

Recursion theory on orderings. II

1980 ◽  
Vol 45 (2) ◽  
pp. 317-333 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Large Class ◽  
General Model ◽  
Recursion Theory ◽  
Second Order ◽  
Splitting Theorem ◽  
Natural Numbers ◽  
Partial Orderings

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.

Note on T-Minimal Complete Bipartite Graphs

1968 ◽  
Vol 11 (5) ◽  
pp. 729-732 ◽  
Author(s):  
I. Z. Bouwer ◽  
I. Broere
Keyword(s):  
Natural Number ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Different Types ◽  
Complete Bipartite Graphs ◽  
Planar Subgraphs ◽  
Proper Subgraph

The thickness of a graph G is the smallest natural number t such that G is the union of t planar subgraphs. A graph G is t-minimal if its thickness is t and if every proper subgraph of G has thickness < t. (These terms were introduced by Tutte in [3]. In [1, p. 51] Beineke employs the term t-critical instead of t-minimal.) The complete bipartite graph K(m, n) consists of m 'dark1 points, n 'light' points, and the mn lines joining points of different types.

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