### Conservative extensions and the jump of a structure

**Soskova, A ***(Sofia University)*

Thursday 07 June 2012, 16:30-17:00

Seminar Room 1, Newton Institute

#### Abstract

The degree spectrum of an abstract structure is a measure of its
complexity. We consider a relation between abstract structures
$\mathfrak{A}$ and $\mathfrak{B}$, possibly with different
signatures and $|\mathfrak{A}|\subseteq |\mathfrak{B}|$, called
conservative extension. We give a characterization of this
relation in terms of definability by computable $\Sigma_n$
formulae on these structures. We show that this relation provides
a finer complexity measure than the one given by degree spectra.
As an application, we receive that the $n$-th jump of a structure
and its Marker's extension are conservative extensions of the
original structure. We present a jump inversion theorem for
abstract structures. We prove that for every natural numbers $n$
and $k$ and each complex enough structure $\mathfrak{A}$, there
is a structure $\mathfrak{B}$, such that the definable by
computable $\Sigma^c_n$ formulae sets on $\mathfrak{A}$ are
exactly the definable by computable $\Sigma^c_k$ formulae on
$\mathfrak{B}$.

#### Presentation

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