A combinatorial property of the homomorphism relation between countable order types

1979 ◽  
Vol 44 (3) ◽  
pp. 403-411 ◽  
Author(s):  
Charles Landraitis
Keyword(s):  
Homomorphic Image ◽  
Linear Order ◽  
Analogous Result ◽  
Infinite Sequence ◽  
Order Type ◽  
Combinatorial Property ◽  
Negative Side ◽  
Order Types

The relation ≪ (is a homomorphic image of) between (linear) order types has properties similar to those of the better known relation ≤ (is embeddable in). For example, the order type η of the rationals not only embeds every countable order type but also maps homomorphically onto . If is scattered, then can be embedded in (ω* + ω)α for some α < ω1. In that case, is also a homomorphic image of (ω* + ω)α [Lan 2]. If is uncountable, then for some uncountable ordinal α, α ≪, α* ≪, or η ≪. Proofs of these facts are much the same for ≤ and ≪.The main theorem of [Lav 1] implies that the embedding relation better-quasiorders the set of countable order types. Our main theorem (§3) states the analogous result for the homomorphism relation. As a consequence, if 0, 1, … is an infinite sequence of countable order types, then there are i, j, i < j, such that i, is a homomorphic image of j. We observed in [Lan 1] that if this is true, then for each countable order type there is a sentence of Lω1ω such that if is a countable order type, then satisfies if and only if is a homomorphic image of . In fact, the motivation for the work leading to this paper came from this observation.On the negative side, it is pointed out (§3) that our theorem cannot be extended as far as that of [Lav 1].

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

scholarly journals Rigidity in order-types

1977 ◽  
Vol 24 (2) ◽  
pp. 203-215 ◽  
Author(s):  
J. L. Hickman
Keyword(s):  
Ordered Set ◽  
Characterization Theorem ◽  
Decomposition Theorem ◽  
Order Type ◽  
Partial Ordering ◽  
Limit Laws ◽  
Order Types ◽  
Totally Ordered Set ◽  
The Axiom Of Choice ◽  
Increasing Sequences

AbstractA totally ordered set (and corresponding order-type) is said to be rigid if it is not similar to any proper initial segment of itself. The class of rigid ordertypes is closed under addition and multiplication, satisfies both cancellation laws from the left, and admits a partial ordering that is an extension of the ordering of the ordinals. Under this ordering, limits of increasing sequences of rigid order-types are well defined, rigid and satisfy the usual limit laws concerning addition and multiplication. A decomposition theorem is obtained, and is used to prove a characterization theorem on rigid order-types that are additively prime. Wherever possible, use of the Axiom of Choice is eschewed, and theorems whose proofs depend upon Choice are marked.

On order-types of models

1972 ◽  
Vol 37 (1) ◽  
pp. 69-70 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Order Type ◽  
Linear Ordering ◽  
Infinite Field ◽  
Strong Limit ◽  
First Order ◽  
Order Types ◽  
Order Language ◽  
Language L ◽  
New Feature

Let T be a theory in a first-order language L. Let L have a predicate ν0 ≺ ν1 such that in every model of T, the interpretation of ≺ is a linear ordering with infinite field. The order-type of this ordering will be called the order-type of the model .Several recent theorems have the following form: if T has a model of order-type ξ then T has a model of order-type ζ (see [1]). We shall add one to the list. The new feature of our result is that the order-type ζ may be in a sense “opposite” to ξ. Silver's Theorem 2.24 of [3] is a corollary of Theorem 1 below.Theorem 1. Let κ be a strong limit number (i.e. μ < κ implies 2μ < κ). Suppose λ < κ, and suppose that for every cardinal μ < κ, T has a model with where the order-type of contains no descending well-ordered sequences of length λ. Then for every cardinal μ ≥ the cardinality ∣L∣ of the language L, T has models and such that(a) the field of is the union of ≤ ∣L∣ well-ordered (inversely well-ordered) parts;(b) .The proof is by Ehrenfeucht-Mostowski models; we presuppose [2].

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

Weakly Homogeneous Order Types

1975 ◽  
Vol 18 (2) ◽  
pp. 159-161 ◽  
Author(s):  
M. E. Adams
Keyword(s):  
Type A ◽  
Order Type ◽  
Order Types

An order type α is said to be weakly homogeneous (ℵ0 homogeneous) if for any x1 < x2 and y1 < y2 there exists an order preserving bijection f on α such that f(xi)= y i for i = 1, 2. The reverse of an order type a is denoted, as usual, by α*. We say that α is order invertible if α*≤α. J. Q. Longyear [5] has asked whether for a weakly homogeneous order type α such that no (non-trivial) interval of α is order invertible we may deduce that every interval of α contains a copy of ηω1 or (ηω1)*.

Decidable discrete linear orders

1988 ◽  
Vol 53 (2) ◽  
pp. 531-539
Author(s):  
M. Moses
Keyword(s):  
Order Type ◽  
Linear Orders ◽  
Order Types

AbstractThree classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.

A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings

1984 ◽  
Vol 49 (2) ◽  
pp. 563-569 ◽  
Author(s):  
Richard Watnick
Keyword(s):  
Ordered Set ◽  
Order Type ◽  
Test Case ◽  
Linear Orderings ◽  
Order Types ◽  
Concrete Model ◽  
Linearly Ordered Set ◽  
Effective Translation ◽  
Recursive Set

An effective translation of the fact that any infinite ordered set contains an infinite ascending or descending sequence is that any infinite recursive set A ⊆ Q has a recursive subset with order type ω or ω*. Tennenbaum's theorem states that this translation is false, and there is a counterexample A with order type ω + ω*. Tennenbaum suggested that this counterexample is an infinite recursive linearly ordered set which is effectively finite, and that the collection of all such counterexamples could provide a concrete model of nonstandard arithmetic. The purpose of this paper is to determine the collection of order types for which there is a counterexample.It is readily seen that any counterexample to the effective translation must have order type ω + Z · α + ω* for some α [2], [3]. Let be the collection of order types α for which there is a counterexample. As a test case, we have previously shown that contains the constructive scattered orderings [3], [4]. In this paper we determine exactly which order types are in . We easily show that if ω+ Z · α + ω* is recursive, then α is . The main result is that . Consequently, .

The Distribution of Subject and Object Agreement and Word Order Type

1996 ◽  
Vol 20 (1) ◽  
pp. 115-161 ◽  
Author(s):  
Anna Siewierska ◽  
Dik Bakker
Keyword(s):  
Word Order ◽  
Order Type ◽  
Morphological Form ◽  
Diachronic Syntax ◽  
Order Types ◽  
Subject And Object ◽  
Object Agreement ◽  
Functional Explanations ◽  
The Impact ◽  
The Relationship

The article examines the distribution and formal realization of Subject and Object agreement markers in different word order types on the basis of a sample of 237 languages. Special attention is paid to the genetic and areal stratification of agreement markers and the impact of these two parameters on the relationship between agreement and word order type emerging from this investigation as opposed to those of previous studies, especially that of Hawkins & Gilligan (1988) and Nichols (1992). The relationship between agreement and word order type is considered in the light of the currently entertained functional explanations for the presence of agreement which are put into question by the high incidence of agreement in V3 languages. The formal realization of the agreement markers (their morphological form and also order relative to each other) in different word order types is investigated relative to the Universal Suffixing Preference, the Head Ordering Principle and the Diachronic Syntax Hypothesis. It is argued that though due to genetic and areal differences in the formal realization of agreement markers, none of the above three hypotheses concerning the relationship between the formal realization of affixal morphemes and word order type provide an adequate account of the cross-linguistic data, the Diachronic Syntax Hypothesis fares better than the other two, particularly in regard to the formal reflexes of object agreement markers. By comparing the results stemming from our sample with those of other samples we seek to draw attention to how areal biases in samples may affect cross-linguistic generalizations. In doing so we hope to highlight the need for developing a sound sampling methodology.

scholarly journals An Optimal Algorithm for Reconstructing Point Set Order Types from Radial Orderings

2017 ◽  
Vol 27 (01n02) ◽  
pp. 57-83 ◽  
Author(s):  
Oswin Aichholzer ◽  
Vincent Kusters ◽  
Wolfgang Mulzer ◽  
Alexander Pilz ◽  
Manuel Wettstein
Keyword(s):  
Polynomial Time ◽  
Processing Time ◽  
Optimal Algorithm ◽  
Order Type ◽  
Convex Hulls ◽  
Radial System ◽  
Polynomial Time Algorithms ◽  
Order Types ◽  
Additional Processing ◽  
Point Set

Let [Formula: see text] be a set of [Formula: see text] labeled points in the plane. The radial system of [Formula: see text] describes, for each [Formula: see text], the order in which a ray that rotates around [Formula: see text] encounters the points in [Formula: see text]. This notion is related to the order type of [Formula: see text], which describes the orientation (clockwise or counterclockwise) of every ordered triple in [Formula: see text]. Given only the order type, the radial system is uniquely determined and can easily be obtained. The converse, however, is not true. Indeed, let [Formula: see text] be the radial system of [Formula: see text], and let [Formula: see text] be the set of all order types with radial system [Formula: see text] (we define [Formula: see text] for the case that [Formula: see text] is not a valid radial system). Aichholzer et al. (Reconstructing Point Set Order Types from Radial Orderings, in Proc. ISAAC 2014) show that [Formula: see text] may contain up to [Formula: see text] order types. They also provide polynomial-time algorithms to compute [Formula: see text] when only [Formula: see text] is given. We describe a new algorithm for finding [Formula: see text]. The algorithm constructs the convex hulls of all possible point sets with the radial system [Formula: see text]. After that, orientation queries on point triples can be answered in constant time. A representation of this set of convex hulls can be found in [Formula: see text] queries to the radial system, using [Formula: see text] additional processing time. This is optimal. Our results also generalize to abstract order types.

Linear Order Types of Nonrecursive Presentability

1985 ◽  
Vol 31 (31-34) ◽  
pp. 495-501 ◽  
Author(s):  
Dev Kumar Roy
Keyword(s):  
Linear Order ◽  
Order Types
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