Dense Linear Order Research Articles

Asymmetric regular types

We study asymmetric regular global types p∈S1(C). If p is regular and A-asymmetric then there exists a strict order such that Morley sequences in p over A are strictly increasing (we allow Morley sequences to be indexed by elements of a linear order). We prove that for any small model M⊇A maximal Morley sequences in p over A consisting of elements of M have the same (linear) order type, denoted by Invp,A(M). In the countable case we determine all possibilities for Invp,A(M): either it can be any countable linear order, or in any M⊇A it is a dense linear order (provided that it has at least two elements). Then we study relationship between Invp,A(M) and Invq,A(M) when p and q are strongly regular, A-asymmetric, and such that p↾A and q↾A are not weakly orthogonal. We distinguish two kinds of non-orthogonality: bounded and unbounded. In the bounded case we prove that Invp,A(M) and Invq,A(M) are either isomorphic or anti-isomorphic. In the unbounded case, Invp,A(M) and Invq,A(M) may have distinct cardinalities but we prove that their Dedekind completions are either isomorphic or anti-isomorphic. We provide examples of all four situations.

Tame properties of sets and functions definable in weakly o-minimal structures

Let \({{\mathcal{M}}=(M, <, \ldots )}\) be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in \({{\mathcal{M}}}\) which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in \({{\mathcal{M}}}\) satisfy an extended version of IVP. After introducing a weak version of definable connectedness in \({{\mathcal{M}}}\) , we prove that strong cells in \({{\mathcal{M}}}\) are weakly definably connected, so every set definable in \({{\mathcal{M}}}\) is a finite union of its weakly definably connected components, provided that \({{\mathcal{M}}}\) has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in \({{\mathcal{M}}}\) and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph (NDG) which was examined for definable functions in (Dolich et al. in Trans. Am. Math. Soc. 362:1371–1411, 2010) and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure \({{\mathcal{M}}}\) having cell decomposition, satisfies NDG, i.e. every definable function in \({{\mathcal{M}}}\) has no dense graph.

Dependent $T$ and existence of limit models

Does the class of linear orders have (one of the variants of) the so called $(\lambda,\kappa)$-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive results. More generally, letting $T$ be a complete first order theory and for simplicity assume G.C.H., for regular $\lambda > \kappa > |T|$ does $T$ have (variants of) a $(\lambda,\kappa)$-limit models, except for stable $T$? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent $T$ we get negative results. We deal more with linear orders.

The dark side of interval temporal logic: marking the undecidability border

Unlike the Moon, the dark side of interval temporal logics is the one we usually see: their ubiquitous undecidability. Identifying minimal undecidable interval logics is thus a natural and important issue in that research area. In this paper, we identify several new minimal undecidable logics amongst the fragments of Halpern and Shoham's logic HS, including the logic of the overlaps relation, over the classes of all finite linear orders and all linear orders, as well as the logic of the meets and subinterval relations, over the classes of all and dense linear orders. Together with previous undecidability results, this work contributes to bringing the identification of the dark side of interval temporal logics very close to the definitive picture.

Extensions of ordered theories by generic predicates

Given a theoryTextending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {&lt;}, we are interested in extensionsT′ ofTin languages extending ℒ by unary relation symbols that are each interpreted in models ofT′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namelyT= Th(〈ℝ, &lt;, +, ·〉) andT′ = Th(〈ℝ, &lt;, +, · ℚ 〉). Recall thatTis o-minimal, and so every open set definable in any model ofThas only finitely many definably connected components. But it is well known that 〈ℝ, &lt;, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models ofTare essentially as simple as possible, whileT′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalgis the set of real algebraic numbers andT′ Th(〈ℝ, &lt;, +, ·, 〈alg〉), then no model ofT′ defines any open set (of any arity) that is not definable in the underlying model ofT.

A scattering of orders

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $ \mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $ \mathcal B$. More generally, we say that a partial ordering is $ \kappa $-scattered if it does not contain a copy of any $ \kappa $-dense linear ordering. We prove analogues of Hausdorff's result for $ \kappa $-scattered linear orderings, and for $ \kappa $-scattered partial orderings satisfying the finite antichain condition. We also study the $ \mathbb{Q}_\kappa $-scattered partial orderings, where $ \mathbb{Q}_\kappa $ is the saturated linear ordering of cardinality $ \kappa $, and a partial ordering is $ \mathbb{Q}_\kappa $-scattered when it embeds no copy of $ \mathbb{Q}_\kappa $. We classify the $ \mathbb{Q}_\kappa $-scattered partial orderings with the finite antichain condition relative to the $ \mathbb{Q}_\kappa $-scattered linear orderings. We show that in general the property of being a $ \mathbb{Q}_\kappa $-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions.

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ∞, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand–Raikov theorem holds for topological subgroups of S ∞: for all such groups, continuous irreducible representations separate points in the group.

Notes on the Model Theory of DeMorgan Logics

We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Priest's Collapsing Lemma and Dunn's Theorem in 3-Valued Logic.

Expressing cardinality quantifiers in monadic second-order logic over chains

AbstractWe investigate the extension of monadic second-order logic of order with cardinality quantifiers “there exists uncountably many sets such that…” and “there exists continuum many sets such that … ”. We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders.

Spectra of highn and non-lown degrees

We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of non-lown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order ℚ. New results include realizations of the set of non-lown Turing degrees as the spectrum of a relation on ℚ for all n≥1, and a realization of the set of all non-lown Turing degrees as the spectrum of a linear order whenever n≥2. The state of current knowledge is summarized in a table in the concluding section.

Linear Quantifier Elimination

This paper presents verified quantifier elimination procedures for dense linear orders (two of them novel), for real and for integer linear arithmetic. All procedures are defined and verified in th...

Σ-definability of uncountable models of c-simple theories

We show that each c-simple theory with an additional discreteness condition has an uncountable model Σ-definable in ℍ\( \mathbb{H} \)(\( \mathbb{L} \)), where \( \mathbb{L} \) is a dense linear order. From this we establish the same for all c-simple theories of finite signature that are submodel complete.

The complexity of temporal constraint satisfaction problems

A temporal constraint language is a set of relations that has a first-order definition in(Q;&lt;), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint languages, then the CSP can be solved in polynomial time; otherwise, the CSP is NP-complete. Our proof combines model-theoretic concepts with techniques from universal algebra, and also applies the so-called product Ramsey theorem, which we believe will useful in similar contexts of constraint satisfaction complexity classification. An extended abstract of this article appeared in the proceedings of STOC'08.

Structures having o-minimal open core

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R \mathfrak R be an expansion of a densely ordered group ( R , &gt; , ∗ ) (R,&gt;,*) that is definably complete and satisfies the uniform finiteness property. Then the open core of R \mathfrak R is o-minimal. Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.

A partition theorem for a large dense linear order

Let κ be a cardinal which is measurable after generically adding ℶκ+ω many Cohen subsets to κ, and let ℚκ = (Q, ≤ Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t + such that the set [Q] m of m-tuples from Q can be partitioned into classes 〈C i : i < t + }〉 such that for any coloring a class C i in fewer than κ colors, there is a copy ℚ* of ℚκ such that [ℚ*] m ⋂ C i is monochromatic. It follows that $$ \mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m $$ , that is, for any coloring of [ℚκ] m with fewer than κ colors there is a copy Q′ ⊆ Q of ℚκ such that [Q′] m has at most t + colors. On the other hand, we show that there are colorings of ℚκ such that if Q′ ⊆ Q is any copy of ℚκ then C i ⋂ [Q′] ≠ ø; for all i < t + , and hence $$ \mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m $$ . We characterize t + as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t 2 + = 2 and for m > 2 we have t + > t m , the m-th tangent number. The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.

Complete and Terminating Tableau for the Logic of Proper Subinterval Structures Over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work.

Tableaux for Logics of Subinterval Structures over Dense Orderings

In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.

Resplendent models and $${\Sigma_1^1}$$ -definability with an oracle

In this article we find some sufficient and some necessary \({\Sigma^1_1}\) -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal arguments are used in conjunction with Pabion’s theorem that ensures that certain oracles are coded in a sufficiently saturated model of arithmetic. Examples of applications are provided for the theories of dense linear orders and of discrete linear orders. These results are then generalised to other ω-categorical theories and theories with a unique countable recursively saturated model.

A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the relationship between size and width cannot be made subpolynomial.

Spectra of structures and relations

AbstractWe consider embeddings of structures which preserve spectra: if g : ℳ → with computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on ). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on . Finally, we consider the extent to which all spectra of unary relations on the structure ℒ may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c′ ≥ τ 0″.