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Sibirsk. Mat. Zh., 2009, Volume 50, Number 3, Pages 625–630 (Mi smj1987)  

This article is cited in 1 scientific paper (total in 1 paper)

On expansions and extensions of powerful digraphs

S. V. Sudoplatov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph $\exists$-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list $p_1(x),…,p_n(x)$ of all nonprincipal 1-types in an LCC theory, if all types $r(x_1,…,x_m)$ containing $p_{i_1}(x_1)\cup\cdots\cup p_{i_m}(x_m)$ are dominated by some type $q$ then $q$ is a powerful type.

Keywords: powerful digraph, powerful type, type-stable theory, Ehrenfeucht theory, locally countably categorical theory.

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English version:
Siberian Mathematical Journal, 2009, 50:3, 498–502

Bibliographic databases:

UDC: 510.67
Received: 22.05.2007

Citation: S. V. Sudoplatov, “On expansions and extensions of powerful digraphs”, Sibirsk. Mat. Zh., 50:3 (2009), 625–630; Siberian Math. J., 50:3 (2009), 498–502

Citation in format AMSBIB
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\by S.~V.~Sudoplatov
\paper On expansions and extensions of powerful digraphs
\jour Sibirsk. Mat. Zh.
\yr 2009
\vol 50
\issue 3
\pages 625--630
\mathnet{http://mi.mathnet.ru/smj1987}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2555887}
\transl
\jour Siberian Math. J.
\yr 2009
\vol 50
\issue 3
\pages 498--502
\crossref{https://doi.org/10.1007/s11202-009-0056-x}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-67650503193}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. B. Sh. Kulpeshov, S. V. Sudoplatov, “Linearly Ordered Theories which are Nearly Countably Categorical”, Math. Notes, 101:3 (2017), 475–483  mathnet  crossref  crossref  mathscinet  isi  elib
  • Сибирский математический журнал Siberian Mathematical Journal
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