Title: $\Sigma$-Definability in Hereditary Finite Superstructures and Pairs of Models
Abstract: 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.
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