Countable Order Research Articles

Generalized metric properties on hyperspaces with the Vietoris topology

We present interrelations between several topological properties of, for example, spaces with a point-countable base, spaces with a σ-hereditarily closure-preserving k-network, spaces with a base of countable order, spaces with a point-regular base, Nagata-spaces, c-semi-stratifiable spaces, γ-spaces, or semi-metrizable spaces of X and the hyperspaces of nonempty compact subsets and finite subsets with the Vietoris topology.

THE POLARISED PARTITION RELATION FOR ORDER TYPES

AbstractWe analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

Posets of Copies of Countable Non-Scattered Labeled Linear Orders

We show that the poset of copies \(\mathbb {P} (\mathbb {Q}_{n} )=\langle \{ f[X]: f\in \text {Emb} (\mathbb {Q}_{n} ) \},\subset \rangle \) of the countable homogeneous universal n-labeled linear order, \(\mathbb {Q}_{n}\), is forcing equivalent to the poset \(\mathbb {S} \ast \pi \), where \(\mathbb {S}\) is the Sacks perfect set forcing and \(1_{\mathbb {S}} \Vdash `` \pi \) is an atomless separative σ-closed forcing”. Under CH (or under some weaker assumptions) \(1_{\mathbb {S}} \Vdash `` \pi \) is forcing equivalent to P(ω)/Fin”. In addition, these statements hold for each countable non-scattered n-labeled linear order \(\mathbb {L}\) and we have \(\text {ro} \text {sq} \mathbb {P} (\mathbb {L} )\cong \text {ro} \text {sq} \mathbb {P} (\mathbb {Q}_{n} )\cong \text {ro} \text {sq} (\mathbb {S} \ast \pi )\).

Hyperresolution for Gödel logic with truth constants

In the paper, we generalise the well-known hyperresolution principle to the general first-order Gödel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Gödel logic by intermediate truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators. We solve the deduction problem of a formula from a countable theory in this expansion. We expand Gödel logic by a countable set of intermediate truth constants of the form c¯, c∈(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form ε1⋄ε2 where εi is an atom or a truth constant or a quantified atom, and ⋄ is a connective ≖, equality, or ≺, strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0,1], respectively. We shall investigate the canonical standard completeness, where the semantics of Gödel logic is given by the standard G-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.

Metrizability of minimal and unbounded topologies

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and L∞. Minimal topologies connect with the recent, and actively studied, subject of “unbounded convergences”. In fact, a Hausdorff locally solid topology τ on a vector lattice X is minimal iff it is Lebesgue and the τ and unbounded τ-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, uτ, associated to τ on X. Regarding metrizability, we prove that if τ is a locally solid metrizable topology then uτ is metrizable iff there is a countable set A with I(A)‾τ=X. We prove that a minimal topology is metrizable iff X has the countable sup property and a countable order basis. In line with the idea that uo-convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo-convergence that generalize classical relations between convergence almost everywhere and convergence in measure.

Properties of Elementary Embeddability in Model Theory

In the class of models of some theory in a countable first order language, we consider relationships between isomorphism, elementary embeddability, and elementary equivalence.

Reverse mathematics and order theoretic fixed point theorems

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within $$\mathsf {RCA_0}$$RCA0 give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over $$\mathsf {RCA_0}$$RCA0. Then we within $$\mathsf {RCA_0}$$RCA0 give proofs of Knaster---Tarski fixed point theorem, Tarski---Kantorovitch fixed point theorem, Bourbaki---Witt fixed point theorem, and Abian---Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian---Brown least fixed point theorem, Davis' converse for countable lattices, Markowski's converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over $$\mathsf {RCA_0}$$RCA0. Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.

Impulsive motion on synchronized spatial temporal grids

We introduce a family of kinetic vector fields on countable space-time grids and study related impulsive second order initial value Cauchy problems. We then construct special examples for which orbits and attractors display unusual analytic and geometric properties.

Epimorphisms Between Linear Orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.

Maximal Chains of Isomorphic Suborders of Countable Ultrahomogeneous Partial Orders

We investigate the poset 〈ℙ(𝕏) ∪ {∅}, ⊂〉, where 𝕏 is a countable ultrahomogeneous partial order and ℙ(𝕏) the set of suborders of 𝕏 isomorphic to 𝕏. For 𝕏 different from (resp. equal to) a countable antichain the order types of maximal chains in 〈ℙ(𝕏) ∪ {∅}, ⊂〉 are characterized as the order types of compact (resp. compact and nowhere dense) sets ofreals having the minimum non-isolated.

Countably categorical quite o-minimal theories

UDC 510.67 We present a complete description of ℵ0-categorical quite o-minimal theories introduced earlier as an extension of the class of o-minimal theories but “quite” inherited various their properties. Bibliography :6 titles. Let L be a countable first order language. Throughout the paper, we consider L-structures and assume that L contains a binary relation symbol < interpreted as a linear ordering in these structures. For arbitrary subsets A and B of a structure M we write A<B if a<b for a ∈ A and b ∈ B. For an arbitrary complete type p we denote by p(M ) the set of realizations of p in M . (∗)

The topological Baumgartner-Hajnal theorem

Two new topological partition relations are proved. These are \begin{equation*} \omega _{1} \to (top \alpha +1)^{2}_{k}\end{equation*} and \begin{equation*} \mathbb {R} \to (top \alpha +1)^{2}_{k}\end{equation*} for all $\alpha < \omega _{1}$ and all $k< \omega$. Here the prefix “top” means that the homogeneous set $\alpha +1$ is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.

The Big Bush machine

In this paper we study an example-machine Bush(S,T) where S and T are disjoint dense subsets of R. We find some topological properties that Bush(S,T) always has, others that it never has, and still others that Bush(S,T) might or might not have, depending upon the choice of the disjoint dense sets S and T. For example, we show that every Bush(S,T) has a point-countable base, is hereditarily paracompact, is a non-Archimedean space, is monotonically ultra-paracompact, is almost base-compact, weakly α-favorable and a Baire space, and is an α-space in the sense of Hodel. We show that Bush(S,T) never has a σ-relatively discrete dense subset (and therefore cannot have a dense metrizable subspace), is never Lindelöf, and never has a σ-disjoint base, a σ-point-finite base, a quasi-development, a Gδ-diagonal, or a base of countable order. We show that Bush(S,T) cannot be a β-space in the sense of Hodel and cannot be a p-space in the sense of Arhangelskii or a Σ-space in the sense of Nagami. We show that Bush(P,Q) is not homeomorphic to Bush(Q,P). Finally, we show that a careful choice of the sets S and T can determine whether the space Bush(S,T) has strong completeness properties such as countable regular co-compactness, countable base compactness, countable subcompactness, and ω-Čech-completeness, and we use those results to find disjoint dense subsets S and T of R, each with cardinality 2ω, such that Bush(S,T) is not homeomorphic to Bush(T,S). We close with a family of questions for further study.

Spaces with sharp bases and with other special bases of countable order

We study spaces with sharp bases and bases of countable order. A characterization of spaces with external bases of countable order is established (Theorem 2.7). Some necessary and sufficient conditions for a space X×S, where S is the convergent sequence, to have a sharp base are given (Theorem 3.2). It follows that a pseudocompact space X is metrizable iff X×S has a sharp base (Corollary 3.3). It is proved that a sharp base of finite rank is a uniform base (Theorem 4.4). Some other new results are also obtained, and some open questions are formulated.

A complicated ω-stable depth 2 theory

AbstractWe present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

The Partial Ordering on the Automorphism Group of the Countable Generic Partial Order

The countable generic poset (P, ≤ ) is the Fraisse limit of the amalgamation class of finite partially ordered sets (see Glass et al., Math Z 214:55–66, 1993; Schmerl, Algebra Univers 9:317–321, 1979). It is homogeneous and \(\aleph_0\)-categorical with quantifier elimination. This paper concerns the structure (G, ∘ , ≤ ), where \((G,\circ)=\textrm{Aut}(P,\leq)\) and ≤ is the pointwise ordering on G. This is a natural structure to look at, because the ordering on G is ∅-definable up to reversal in the language { ∘ } (but this fact is not proved here). In this paper I show that (G, ≤ ) is elementarily equivalent to (P, ≤ ) itself. More generally, (G, ∘ , ≤ ) satisfies a weakening of the existential closure property for partially ordered groups. (Existential closure in groups has been studied for example in Higman and Scott.) This requires one to study the group G ∗ , obtained by freely adjoining a finite set of generators to G.

Model theory without choice? Categoricity

AbstractWe prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α &gt; 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

Weakly continuously Urysohn spaces

We study weakly continuously Urysohn spaces, which were introduced in [P.L. Zenor, Continuously extending partial functions, Proc. Amer. Math. Soc. 135 (1) (2007) 305–312]. We show that every weakly continuously Urysohn wΔ-space has a base of countable order, that separable weakly continuously Urysohn spaces are submetrizable, hence continuously Urysohn, that monotonically normal weakly continuously Urysohn spaces are hereditarily paracompact, and that no linear extension of any uncountable subspace of the Sorgenfrey line is weakly continuously Urysohn. These results generalize various results in the literature concerning continuously Urysohn spaces.

Domain representability and the Choquet game in Moore and BCO-spaces

In this paper we investigate the role of domain representability and Scott-domain representability in the class of Moore spaces and the larger class of spaces with a base of countable order. We show, for example, that in a Moore space, the following are equivalent: domain representability; subcompactness; the existence of a winning strategy for player α (= the nonempty player) in the strong Choquet game Ch ( X ) ; the existence of a stationary winning strategy for player α in Ch ( X ) ; and Rudin completeness. We note that a metacompact Čech-complete Moore space described by Tall is not Scott-domain representable and also give an example of Čech-complete separable Moore space that is not co-compact and hence not Scott-domain representable. We conclude with a list of open questions.

The Length of Chains in Modular Algebraic Lattices

We show that, for a large class of countable order types α, a modular algebraic lattice L contains no chain of type α if and only if K(L), the join-semilattice of compact elements of L, contains neither a chain of type α nor a join-subsemilattice isomorphic to [ω] < ω , the set of finite subsets of ω ordered by inclusion. We give a description of the indivisible members of this class. It includes the order types ω * of the chain of negative integers and η of the chain of rational numbers.