Authors: Moconja, Slavko
Tanović, Predrag 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Asymmetric regular types
Journal: Annals of Pure and Applied Logic
Volume: 166
Issue: 2
First page: 93
Last page: 120
Issue Date: 1-Jan-2015
Rank: M22
ISSN: 0168-0072
DOI: 10.1016/j.apal.2014.09.003
Abstract: 
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.
Keywords: Complete theory | Global type | Invariant type | Linear order | Morley sequence | Regular type
Publisher: Elsevier
Project: Algebraic, logical and combinatorial methods with applications in theoretical computer science 
Representations of logical structures and formal languages and their application in computing 

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