An ordered structure of rank two related to Dulac's Problem

A. Dolich; P. Speissegger

Displaying similar documents to “An ordered structure of rank two related to Dulac's Problem”

Relatively complete ordered fields without integer parts

Mojtaba Moniri, Jafar S. Eivazloo (2003)

Fundamenta Mathematicae

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We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series $[[F^{G}]]$ with exponents in a totally ordered Abelian group G and coefficients in an ordered field F. This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that $[[F^{G}]]$ is always Scott complete. In contrast, the Puiseux...

Cycle-pancyclism in bipartite tournaments I

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k <...

Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

Amitayu Banerjee, Zalán Gyenis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size $ℵ_{α}$ , then the set has size $ℵ_{α}$ for any regular $ℵ_{α}$ . $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. $\circ$ Every partially...

Cycle-pancyclism in bipartite tournaments II

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was...

On the jump number of lexicographic sums of ordered sets

Hyung Chan Jung, Jeh Gwon Lee (2003)

Czechoslovak Mathematical Journal

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Let $Q$ be the lexicographic sum of finite ordered sets $Q_{x}$ over a finite ordered set $P$ . For some $P$ we can give a formula for the jump number of $Q$ in terms of the jump numbers of $Q_{x}$ and $P$ , that is, $s (Q) = s (P) + \sum_{x \in P} s (Q_{x})$ , where $s (X)$ denotes the jump number of an ordered set $X$ . We first show that $w (P) - 1 + \sum_{x \in P} s (Q_{x}) \leq s (Q) \leq s (P) + \sum_{x \in P} s (Q_{x})$ , where $w (X)$ denotes the width of an ordered set $X$ . Consequently, if $P$ is a Dilworth ordered set, that is, $s (P) = w (P) - 1$ , then the formula holds. We also show that it holds again if $P$ is bipartite. Finally, we prove that the lexicographic...

Embedding orders into the cardinals with $D C_{κ}$

Asaf Karagila (2014)

Fundamenta Mathematicae

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Jech proved that every partially ordered set can be embedded into the cardinals of some model of ZF. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $Z F + D C_{< κ}$ for any regular κ. We use this theorem to show that for all κ, the assumption of $D C_{κ}$ does not entail that there are no decreasing chains of cardinals. We also show how to extend the result to and embed into the cardinals a proper class which is definable over the ground model....

Spaces of continuous step functions over LOTS

Raushan Z. Buzyakova (2006)

Fundamenta Mathematicae

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We investigate spaces $C_{p} (\cdot, n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_{p} (\cdot, n)$ over LOTS. We also characterize countably compact LOTS whose $C_{p} (\cdot, n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

Minimal and minimum size latin bitrades of each genus

James Lefevre, Diane Donovan, Nicholas J. Cavenagh, Aleš Drápal (2007)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that $T^{\circ}$ and $T^{☆}$ are partial latin squares of order $n$ , with the property that each row and each column of $T^{\circ}$ contains the same set of entries as the corresponding row or column of $T^{☆}$ . In addition, suppose that each cell in $T^{\circ}$ contains an entry if and only if the corresponding cell in $T^{☆}$ contains an entry, and these entries (if they exist) are different. Then the pair $T = (T^{\circ}, T^{☆})$ forms a . The of $T$ is the total number of filled cells in $T^{\circ}$ (equivalently $T^{☆}$ ). The latin bitrade is if there is no...

Linear forests and ordered cycles

Guantao Chen, Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson, Linda Lesniak, Florian Pfender (2004)

Discussiones Mathematicae Graph Theory

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A collection $L = P ¹ \cup P ² \cup . . . \cup P^{t}$ (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian. ...

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

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We prove that every vertex v of a tournament T belongs to at least $m a x m i n δ ⁺ (T), 2 δ ⁺ (T) - d ⁺_{T} (v) + 1, m i n δ ¯ (T), 2 δ ¯ (T) - d ¯_{T} (v) + 1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d ⁺_{T} (v)$ (or $d ¯_{T} (v)$ ) is the out-degree (resp. in-degree) of v.

Pairs of forbidden class of subgraphs concerning $K_{1, 3}$ and P₆ to have a cycle containing specified vertices

Takeshi Sugiyama, Masao Tsugaki (2009)

Discussiones Mathematicae Graph Theory

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In [3], Faudree and Gould showed that if a 2-connected graph contains no $K_{1, 3}$ and P₆ as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair $K_{1, 3}$ and P₆, and prove that the forbidden families implies the existence of cycles passing through specified vertices.

Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

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For any integer $n > 1$ , we provide a parametric family of biquadratic fields with class groups having $n$ -rank at least 2. Moreover, in some cases, the $n$ -rank is bigger than 4.

On $p^{2}$ -Ranks in the Class Field Tower Problem

Christian Maire, Cam McLeman (2014)

Annales mathématiques Blaise Pascal

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Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^{2}$ -rank of the class group as a quantity of relevance in the $p$ -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class...

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

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Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD,...

Counting linearly ordered spaces

Gerald Kuba (2014)

Colloquium Mathematicae

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For a transfinite cardinal κ and i ∈ 0,1,2 let $ℒ_{i} (κ)$ be the class of all linearly ordered spaces X of size κ such that X is totally disconnected when i = 0, the topology of X is generated by a dense linear ordering of X when i = 1, and X is compact when i = 2. Thus every space in ℒ₁(κ) ∩ ℒ₂(κ) is connected and hence ℒ₁(κ) ∩ ℒ₂(κ) = ∅ if $κ < 2^{ℵ ₀}$ , and ℒ₀(κ) ∩ ℒ₁(κ) ∩ ℒ₂(κ) = ∅ for arbitrary κ. All spaces in ℒ₁(ℵ₀) are homeomorphic, while ℒ₂(ℵ₀) contains precisely ℵ₁ spaces up to homeomorphism. The...