On the quasi-ordering of Borel linear orders under embeddability

1990 ◽  
Vol 55 (2) ◽  
pp. 537-560 ◽  
Author(s):  
Alain Louveau ◽  
Jean Saint-Raymond
Keyword(s):  
Linear Orders ◽  
Lexicographic Ordering ◽  
Quasi Ordering

AbstractWe provide partial answers to the following problem: Is the class of Borel linear orders well-quasi-ordered under embeddability? We show that it is indeed the case for those Borel orders which are embeddable in Rω, with the lexicographic ordering. For Borel orders embeddable in R2, our proof works in ZFC, but it uses projective determinacy for Borel orders embeddable in some Rn, n < ω, and hyperprojective determinacy for the general case.

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.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

scholarly journals FINDING DESCENDING SEQUENCES THROUGH ILL-FOUNDED LINEAR ORDERS

2021 ◽  
pp. 1-39
Author(s):  
JUN LE GOH ◽  
ARNO PAULY ◽  
MANLIO VALENTI
Keyword(s):  
Linear Orders

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

Well-quasi-ordering hereditarily finite sets

2013 ◽  
Vol 90 (6) ◽  
pp. 1278-1291 ◽  
Author(s):  
Alberto Policriti ◽  
Alexandru I. Tomescu
Keyword(s):  
Finite Sets ◽  
Quasi Ordering

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

2014 ◽  
Vol 79 (4) ◽  
pp. 1001-1019 ◽  
Author(s):  
ASHER M. KACH ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Direct Product ◽  
Disjoint Union ◽  
Second Order ◽  
Linear Orders ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Natural Classes ◽  
Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

scholarly journals On the Ungrammaticality of Remnant Movement in the Derivation of Greenberg's Universal 20

2011 ◽  
Vol 42 (3) ◽  
pp. 445-469 ◽  
Author(s):  
Sam Steddy ◽  
Vieri Samek-Lodovici
Keyword(s):  
Optimality Theory ◽  
Linear Orders ◽  
Original Analysis ◽  
Remnant Movement

We propose an analysis that derives Cinque's (2005) typology of linear orders involving a demonstrative, numeral, adjective, and noun through four Optimality Theory constraints requiring leftward alignment of these items. We show that remnant movement is ungrammatical whenever it produces universally suboptimal alignments, compared with remnant-movement-free structures. Any movement is permitted, but only the best alignment configurations surface as grammatical. We also show that Cinque's original analysis must encode the structural derivations of all attested orders as parametric values of the associated languages. Our analysis need not make similar structural stipulations, as the different attested structures emerge from constraint reranking.

On the Expressiveness of the Interval Logic of Allen’s Relations Over Finite and Discrete Linear Orders

2014 ◽  
pp. 267-281 ◽  
Author(s):  
Luca Aceto ◽  
Dario Della Monica ◽  
Anna Ingólfsdóttir ◽  
Angelo Montanari ◽  
Guido Sciavicco
Keyword(s):  
Linear Orders ◽  
Interval Logic

scholarly journals On Well-quasi-ordering Finite Sequences

1989 ◽  
Vol 10 (3) ◽  
pp. 227-230 ◽  
Author(s):  
Ulrich Bollerhoff
Keyword(s):  
Finite Sequences ◽  
Quasi Ordering
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