Countable Order Research Articles

Measure theory aspects of locally countable orderings

AbstractWe prove that for any locally countable partial order ℙ = (2ε, ≤p, there exists a nonmeasurable antichain in ℙ. Some applications of the result are also presented.

Compact [formula omitted] sets

In this paper we study spaces in which each compact subset is a G δ -set and compare them to H.W. Martin's c-semi-stratifiable (CSS) spaces, i.e. spaces in which compact sets are G δ -sets in a uniform way. We prove that a (countably) compact subset of a Hausdorff space X is metrizable and a G δ -subset of X provided X has a δθ-base, or a point-countable, T 1 -point-separating open cover, or a quasi- G δ -diagonal. We also show that any compact subset of a Hausdorff space X having a base of countable order must be a G δ -subset of X and note that this result does not hold for countably compact subsets of BCO-spaces. We characterize CSS spaces in terms of certain functions g ( n , x ) and prove a “local implies global” theorem for submetacompact spaces that are locally CSS. In addition, we give examples showing that even though every compact subset of a space with a point-countable base (respectively, of a space with a base of countable order) must be a G δ -set, there are examples of such spaces that are not CSS. In the paper's final section, we examine the role of the CSS property in the class of generalized ordered (GO) spaces. We use a stationary set argument to show that any monotonically normal CSS space is hereditarily paracompact. We show that, among GO-spaces with σ-closed-discrete dense subsets, being CSS and having a G δ -diagonal are equivalent properties, and we use a Souslin space example due to Heath to show that (consistently) the CSS property is not equivalent to the existence of a G δ -diagonal in the more general class of perfect GO-spaces.

Characterizing an ℵ ∈ -saturated model of superstable NDOP theories by its $$\mathbb{L}_{\infty ,\aleph _ \in ^ - } $$ -theory

Assume a complete countable first order theory is superstable with NDOP. We know that any ℵ ∈ -saturated model of the theory is ℵ ∈ -prime over a non-forking tree of “small” models and its isomorphism type can be characterized by its\(\mathbb{L}_{\infty ,k} \) (dimension qualifiers)-theory, or, if you prefer, appropriate cardinal invariants. We go one step further by providing cardinal invariants which are as finitary as seem reasonable.

C-Quasiminimal Enumeration Degrees

The notion of a C-quasiminimal set, with C an arbitrary subset of the naturals, was introduced by Sasso and presents a relativization of the well-known notion of quasiminimal set which was first constructed by Medvedev for proving the existence of nontotal enumeration degrees. In this article we study the local properties of the partially ordered set of the enumeration degrees containing C-quasiminimal sets. In particular, we prove for arbitrary enumeration degrees c and a that if c<a and a is a total e-degree then each at most countable partial order embeds isomorphically into the partially ordered set of c-quasiminimal e-degrees lying below a.

Uncountable Homogeneous Partial Orders

A partially ordered set (P, ≤) is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of (P, ≤). Assuming the set-theoretic assumption ⋄(ϰ1), it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not (k + 1)-homogeneous. This is impossible in the countable case for k ≥ 4.

The relation of recursive isomorphism for countable structures

AbstractIt is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes.

On Retracts of the Random Graph and Their Natural Order

We prove that the natural order on the idempotents of the endomorphism monoid of the countably infinite random graph is ℵ0-universal; that is, it embeds every countable order. We therefore extend in a strong fashion a result of [2] which showed that the natural order embeds every countable linear order. We consider a refinement of the natural order which embeds every countable quasi-order.

Group Theoretic Properties of the Group of Computable Automorphisms of a Countable Dense Linear Order

We compare Aut(Q), the classical automorphism group of a countable dense linear order, with Autc(Q), the group of all computable automorphisms of such an order. They have a number of similarities, including the facts that every element of each group is a commutator and each group has exactly three nontrivial normal subgroups. However, the standard proofs of these facts in Aut(Q) do not work for Autc(Q). Also, Aut(Q) has three fundamental properties which fail in Autc(Q): it is divisible, every element is a commutator of itself with some other element, and two elements are conjugate if and only if they have isomorphic orbital structures.

Embedding of Countable Orders in Turing Degrees

In their classical papers, Lerman, Lachlan, and Lebeuf developed the embedding method, which provides constructions of initial segments of Turing degrees isomorphic to various partially ordered structures. We analyze this method and prove that there is a nonzero degree below each decreasing chain of degrees uniform in O'. This imposes restrictions on the application of the embedding method.

Some dichotomy theorems for isomorphism relations of countable models

AbstractStrengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

Semi-continuous multifunctions and bases of countable order

This paper is devoted to the problem of selections. An important result in this area is Michael's theorem on double selection for lower semi-continuous closed valued multifunctions. Recently we obtained a generalization of this theorem to a subclass of the so-called generalized metric spaces, namely the class of weakly developable spaces. One of the aims of this paper is to give an extension of our result (hence of Michael's result) to a more general class of spaces, namely the class of spaces with a base of countable order. To do this, we give some results on spaces with a base of countable order which extend those of Wicke and Worrell Jr. Some applications are given. In particular we obtain a criterion of metrizability.

Weak developments and metrization

The notions of a weak k-development and of a weak development, defined in terms of sequences of open covers, were recently introduced by the first and the third authors. The first notion was applied to extend in an interesting way Michael's Theorem on double set-valued selections. The second notion is situated between that of a development and of a base of countable order. To see that a space with a weak development has a base of countable order, we use the classical works of H.H. Wicke and J.M. Worrell. We also introduce and study the new notion of a sharp base, which is strictly weaker than that of a uniform base and strictly stronger than that of a base of countable order and of a weakly uniform base, and which is strongly connected to the notion of a weak development. Several examples are exhibited to prove that the new notions do not coincide with the old ones. In short, our results show that the notions of a weak development and of a sharp base fit very well into already existing system of generalized metrizability properties defined in terms of sequences of open covers or bases. Several open questions are formulated.

The Complexity of the Collection of Countable Linear Orders of the form I + I

AbstractFirst we prove that the set of countable linear orders of the form I + I form a complete analytic set. As a consequence of this we improve a result of Humke and Laczkovich, who showed in [HL] that the set of functions of the form f º f form a true analytic set in C[0,1]. We show that these functions form a complete analytic set, solving a problem mentioned on p. 215 of [Kl] and on p. 4 of [B]

P-Q-Metrizability of Bitopological Spaces with a Dual Base of Countable Order

In this paper the notion of a dual base of countable order for bitopological spaces is introduced, and an analogue of a metrization theorem of A. Arhangel'skii is given.

Abstract structure for Lie pseudogroups

We give the solution of Lie's third fundamental problem for the class of infinite dimensional Lie algebras corresponding to the flat Lie pseudogroups of infinite type. The associated groupoids are infinite dimensional Lie groupoids of the second kind and of countable order.

Preimages of Spaces Having a Base of Countable Order

The main theorem characterizes the quasi‐perfect and perfect preimages of T1‐spaces having a base of countable order. The preimages of T1‐spaces having a base of countable order are connected with the preimages of T1‐spaces having a primitive base under the same type of mappings.

Two subclasses of the class MOBI

Assuming that all spaces are regular we prove that open and compact images of σ-locally compact metric spaces are σ-locally compact metacompact Moore spaces while open and compact images of σ-locally compact metacompact Moore spaces form the class of spaces with a point-countable base of countable order and a closure-preserving closed cover by σ-compact sets. Moreover, this class is the minimal class of regular spaces which contains all σ-locally compact metric spaces and is invariant under open and compact mappings. The complete version of these results gives a characterization of images of C-scattered metric spaces.

Left K‐Completeness in Quasi‐Metric Spaces

AbstractRegular left K‐sequentially complete quasi‐metric spaces are characterized. We deduce that these spaces are complete Aronszajn and that every metrizable space admitting a left K‐sequentially complete quasi‐metric is completely metrizable. We also characterize quasi‐metric spaces having a quasi‐metric left K‐sequential completion in terms of certain bases of countable order.

Elementary embedding between countable Boolean algebras

AbstractFor a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B2 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then ‹MT, ≤› is well-quasi-ordered. ∎ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that B ↾ a is an atomic Boolean algebra and B ↾ s is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that ‹A, &lt; › is partial well-quasi-ordering, it is a partial quasi-ordering and for every {ai, ⃒ i ∈ ω} ⊆ A, there are i &lt; j &lt; ω such that ai ≤ aj. Theorem 2. contains a subset M such that the partial orderings ‹M, ≤ ↾ M› and are isomorphic. ∎ Let M′0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M′0, let B1 ≤′ B2 mean that B1 is embeddable in B2. Remark. ‹M′0, ≤′› is well-quasi-ordered. ∎ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

Theory of Field-Induced Spin Density Wave States in Anisotropic Two-Dimensional Electron Systems

In order to explain recursive cascades of the field-induced spin density wave phase in organic conductors (TMTSF) 2 X discovered recently, a standard model or two-dimensional anisotropic Hubbard model under a perpendicular field is investigated theoretically: The field-induced SDW ground state is in general to be characterized by a set of countable many order parameters (OP) of the Fourier components of the SDW modulation or by a distribution of these OP's. Due to interference between the lattice commensurability and magnetic length the OP distribution differs for every rational field, resulting in a phase diagram with devil's staircase like infinite cascades. The band structure and spatial profile of the FISDW are analyzed and compared with ordinary density wave states.