Weak commutativity of scattered summation of linear orders

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.

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FINDING DESCENDING SEQUENCES THROUGH ILL-FOUNDED LINEAR ORDERS

Journal of Symbolic Logic ◽

10.1017/jsl.2021.15 ◽

2021 ◽

pp. 1-39

Author(s):

JUN LE GOH ◽

ARNO PAULY ◽

MANLIO VALENTI

Keyword(s):

Linear Orders

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Generalized Commutativity of Lattice-Ordered Groups

Mathematica Slovaca ◽

10.1515/ms-2015-0026 ◽

2015 ◽

Vol 65 (2) ◽

Cited By ~ 1

Author(s):

M. R. Darnel ◽

W. C. Holland ◽

H. Pajoohesh

Keyword(s):

Theorem Proving ◽

Natural Generalization ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Weak Commutativity

AbstractIn this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

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A model with Suslin trees but no minimal uncountable linear orders other than ω1 and −ω1

Israel Journal of Mathematics ◽

10.1007/s11856-019-1899-x ◽

2019 ◽

Vol 233 (1) ◽

pp. 199-224

Author(s):

Dániel T. Soukup

Keyword(s):

Linear Orders

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Shuffling of Linear Orders

Canadian Mathematical Bulletin ◽

10.4153/cmb-1995-032-1 ◽

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.

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Two results on borel orders

Journal of Symbolic Logic ◽

10.2307/2274748 ◽

1989 ◽

Vol 54 (3) ◽

pp. 865-874 ◽

Cited By ~ 5

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.

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UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

Journal of Symbolic Logic ◽

10.1017/jsl.2014.54 ◽

2014 ◽

Vol 79 (4) ◽

pp. 1001-1019 ◽

Cited By ~ 2

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.

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

Linguistic Inquiry ◽

10.1162/ling_a_00053 ◽

2011 ◽

Vol 42 (3) ◽

pp. 445-469 ◽

Cited By ~ 12

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.

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