Characterization of posets of intervals

Judita Lihová

Displaying similar documents to “Characterization of posets of intervals”

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...

Orthomodular lattices and closure operations in ordered vector spaces

Jan Florek (2010)

Banach Center Publications

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On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and $A \to A^{⊥ ⊥}$ . It was proved that V is integrally open iff $D (A) = A^{⊥ ⊥}$ for every orthogonal set A ⊆ V. In this paper we generalize this...

On monotone permutations of $ℓ$ -cyclically ordered sets

Ján Jakubík (2006)

Czechoslovak Mathematical Journal

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For an $ℓ$ -cyclically ordered set $M$ with the $ℓ$ -cyclic order $C$ let $P (M)$ be the set of all monotone permutations on $M$ . We define a ternary relation $\bar{C}$ on the set $P (M)$ . Further, we define in a natural way a group operation (denoted by $\cdot$ ) on $P (M)$ . We prove that if the $ℓ$ -cyclic order $C$ is complete and $\bar{C} \neq \emptyset$ , then $(P (M), \cdot, \bar{C})$ is a half cyclically ordered group.

Selfduality of the system of convex subsets of a partially ordered set

Miron Zelina (1993)

Commentationes Mathematicae Universitatis Carolinae

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For a partially ordered set $P$ let us denote by $C o P$ the system of all convex subsets of $P$ . It is found the necessary and sufficient condition (concerning $P$ ) under which $C o P$ (as a partially ordered set) is selfdual.

Generalized M-norms on ordered normed spaces

I. Tzschichholtz, M. R. Weber (2005)

Banach Center Publications

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L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and $m_{\leq}$ -norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and...

A note on weakly $𝔪$ -distributive Boolean algebras

R. H. Lagrange (1970)

Colloquium Mathematicae

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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...

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...

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.

Modifications of the double arrow space and related Banach spaces C(K)

Witold Marciszewski (2008)

Studia Mathematica

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We consider the class of compact spaces $K_{A}$ which are modifications of the well known double arrow space. The space $K_{A}$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_{A}$ spaces and on the isomorphic classification of the Banach spaces $C (K_{A})$ .