Imaginaries in dense pairs of real-closed fields

Rakotonarivo, Tsinjo Odilon (2017-03)


ENGLISH ABSTRACT : Imaginaries are definable equivalence classes, which play an important role in model theory. In this thesis, we are interested in imaginaries of dense pairs of real-closed fields. More precisely, we consider the following problem: is acleq equal to dcleq in dense pairs of real-closed fields? To answer this question, we first present some results about real-closed fields, which are basically completeness, quantifier elimination and elimination of imaginaries. Then, we concentrate on the completeness and near model-completeness for the theory of dense pairs of real-closed fields. And finally, we present the key point of the thesis. Namely, we demonstrate that acleq(∅) = dcleq(∅) but there exists A such that acleq(A) 6= dcleq(A)

AFRIKAANSE OPSOMMING : Imaginêres is definiëerbare ekwivalensieklasse, wat ’n belangrike rol in modelteorie speel. In hierdie tesis stel ons belang in imaginêres in dig pare van reël-geslote liggame. Meer spesifiek beskou ons die volgende probleem: is acleq gelyk aan dcleq in dig pare van reël-geslote liggame? Om hierdie vraag te beantwoord, begin ons met ’n paar resultate oor reëlgeslote liggame, namelik volledigheid, kwantoreliminasie en eliminasie van imaginêres. Daarna behandel ons die volledigheid en byna-modelvolledigheid vir die teorie van dig pare van reël-geslote liggame. Uiteindelik behandel ons die hoofresultat van hierdie tesis, d.w.s. ons bewys dat acleq(∅) = dcleq(∅) maar dat daar A bestaan sodat acleq(A) 6= dcleq(A)

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