$\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models
Algebra i logika, Tome 43 (2004) no. 4, pp. 459-481
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We consider the problem of being $\Sigma$-definable for an uncountable model of a $c$-simple theory in hereditarily finite superstructures over models of another $c$-simple theory. A necessary condition is specified in terms of decidable models and the concept of relative indiscernibility introduced in the paper. A criterion is stated for the uncountable model of a $c$-simple theory to be $\Sigma$-definable in superstructures over dense linear orders, and over infinite models of the empty signature. We prove the existence of a $c$-simple theory (of an infinite signature) every uncountable model of which is not $\Sigma$-definable in superstructures over dense linear orders. Also, a criterion is given for a pair of models to be recursively saturated.
Keywords:
$\Sigma$-definability, $c$-simple theory, model, hereditarily finite superstructure, linear order.
@article{AL_2004_43_4_a4,
author = {A. I. Stukachev},
title = {$\Sigma${-Definability} in {Hereditarily} {Finite} {Superstructures} and {Pairs} of {Models}},
journal = {Algebra i logika},
pages = {459--481},
publisher = {mathdoc},
volume = {43},
number = {4},
year = {2004},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/AL_2004_43_4_a4/}
}
A. I. Stukachev. $\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models. Algebra i logika, Tome 43 (2004) no. 4, pp. 459-481. http://geodesic.mathdoc.fr/item/AL_2004_43_4_a4/