Adding linear orders

2012 ◽  
Vol 77 (2) ◽  
pp. 717-725 ◽  
Author(s):  
Saharon Shelah ◽  
Pierre Simon
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Categorical Theory

AbstractWe address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

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 On cuts in ultraproducts of linear orders I

2016 ◽  
Vol 16 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Mohammad Golshani ◽  
Saharon Shelah
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Regular Cardinals

For an ultrafilter [Formula: see text] on a cardinal [Formula: see text] we wonder for which pair [Formula: see text] of regular cardinals, we have: for any [Formula: see text]-saturated dense linear order [Formula: see text] has a cut of cofinality [Formula: see text] We deal mainly with the case [Formula: see text]

R. e. presented linear orders

1983 ◽  
Vol 48 (2) ◽  
pp. 369-376 ◽  
Author(s):  
Dev Kumar Roy
Keyword(s):  
Linear Order ◽  
Order Relation ◽  
Recursion Theory ◽  
Linear Orders ◽  
Preliminary Results ◽  
Negative Results ◽  
Recursively Enumerable ◽  
Equality Relation ◽  
The Law

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

scholarly journals Finite Homomorphism-Homogeneous Permutations via Edge Colourings of Chains

10.37236/2271 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Igor Dolinka ◽  
Éva Jungábel
Keyword(s):  
Directed Graph ◽  
Linear Order ◽  
Relational Structure ◽  
Traditional View ◽  
Linear Orders ◽  
Finite Set ◽  
Edge Colourings

A relational structure is homomorphism-homogeneous if any homomorphism between its finite substructures extends to an endomorphism of the structure in question. In this note, we characterise all permutations on a finite set enjoying this property. To accomplish this, we switch from the more traditional view of a permutation as a set endowed with two linear orders to a different representation by a single linear order (considered as a directed graph with loops) whose non-loop edges are coloured in two colours, thereby `splitting' the linear order into two posets.

On theories having a finite number of nonisomorphic countable models

1985 ◽  
Vol 50 (3) ◽  
pp. 806-808 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
Finite Number ◽  
Positive Solutions ◽  
Linear Order ◽  
Order Property ◽  
Categorical Theory ◽  
Base Theory ◽  
Almost All ◽  
Countable Models

In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory.Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

Decidability and ℵ0-categoricity of theories of partially ordered sets

1980 ◽  
Vol 45 (3) ◽  
pp. 585-611 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Linear Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Ordered Sets ◽  
Categorical Theory ◽  
Partially Ordered ◽  
Decidable Theory ◽  
Finite Partially Ordered Set

AbstractThis paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following.Corollary 3.3. For every countable ℵ0-categoricalthere is a linear order of A such that (, <) is ℵ0-categorical.Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory.Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory.There is a section dealing just with decidability of partially ordered sets, the main result of this section beingTheorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

scholarly journals An issue based power index

2020 ◽  
Author(s):  
Qianqian Kong ◽  
Hans Peters
Keyword(s):  
Linear Order ◽  
Power Index ◽  
Simple Game ◽  
Power Indices ◽  
Weighted Sum ◽  
Simple Games ◽  
Linear Orders ◽  
Transfer Property ◽  
Power Change ◽  
Equal Power

Abstract An issue game is a combination of a monotonic simple game and an issue profile. An issue profile is a profile of linear orders on the player set, one for each issue within the set of issues: such a linear order is interpreted as the order in which the players will support the issue under consideration. A power index assigns to each player in an issue game a nonnegative number, where these numbers sum up to one. We consider a class of power indices, characterized by weight vectors on the set of issues. A power index in this class assigns to each player the weighted sum of the issues for which that player is pivotal. A player is pivotal for an issue if that player is a pivotal player in the coalition consisting of all players preceding that player in the linear order associated with that issue. We present several axiomatic characterizations of this class of power indices. The first characterization is based on two axioms: one says how power depends on the issues under consideration (Issue Dependence), and the other one concerns the consequences, for power, of splitting players into several new players (no advantageous splitting). The second characterization uses a stronger version of Issue Dependence, and an axiom about symmetric players (Invariance with respect to Symmetric Players). The third characterization is based on a variation on the transfer property for values of simple games (Equal Power Change), besides Invariance with respect to Symmetric Players and another version of Issue Dependence. Finally, we discuss how an issue profile may arise from preferences of players about issues.

scholarly journals Determining Possible and Necessary Winners Given Partial Orders

2011 ◽  
Vol 41 ◽  
pp. 25-67 ◽  
Author(s):  
L. Xia ◽  
V. Conitzer
Keyword(s):  
Linear Order ◽  
Partial Orders ◽  
Scoring Rules ◽  
Linear Orders ◽  
Voting Rules ◽  
Voting Rule

Usually a voting rule requires agents to give their preferences as linear orders. However, in some cases it is impractical for an agent to give a linear order over all the alternatives. It has been suggested to let agents submit partial orders instead. Then, given a voting rule, a profile of partial orders, and an alternative (candidate) c, two important questions arise: first, is it still possible for c to win, and second, is c guaranteed to win? These are the possible winner and necessary winner problems, respectively. Each of these two problems is further divided into two sub-problems: determining whether c is a unique winner (that is, c is the only winner), or determining whether c is a co-winner (that is, c is in the set of winners). We consider the setting where the number of alternatives is unbounded and the votes are unweighted. We completely characterize the complexity of possible/necessary winner problems for the following common voting rules: a class of positional scoring rules (including Borda), Copeland, maximin, Bucklin, ranked pairs, voting trees, and plurality with runoff.

The ⊰-order on submodels

1976 ◽  
Vol 41 (1) ◽  
pp. 215-221
Author(s):  
Leo Marcus
Keyword(s):  
Large Class ◽  
Linear Order ◽  
Partial Ordering ◽  
Natural Order ◽  
Prime Model ◽  
Natural Question ◽  
Categorical Theory ◽  
Elementary Submodels ◽  
Homogeneous Models ◽  
Element Theorem

The relation M1 ⊰ M2, where M1 and M2 are elementary submodels of a given model M, defines a partial ordering ⊰(M). A natural question is: What are the general properties of this ordering and what special properties can it have for given M? Here we give an example of a prime model M for which ⊰(M) is a dense linear order, in particular ⊰(M) is isomorphic to the natural order of (− ∞, ∞], the reals with a last element (Theorem 1). In fact, this is the only way for ⊰(M) to be a linear order when M is prime (Theorem 3).We then remark that if N is a model of an ℵ0-categorical theory then ⊰(N) cannot be a linear order, but we give an example where ⊰(N) is a dense partial order, N a model of an ℵ0-categorical theory.Independently, and about the same time, Benda [1] found nonprime models M where ⊰(M) is isomorphic to any one of a large class of orders, including (− ∞, ∞]. His methods are very similar to ours.Notation. We assume the reader is familar with Vaught [2] where he will find the definitions and standard results pertaining to prime models, homogeneous models, and ℵ0-categorical theories.

Sign in / Sign up

Export Citation Format

Share Document