Browsing by Author "Rakotonarivo, Tsinjo Odilon"

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    The definable (p,q)-theorem for dense pairs of certain geometric structures
    (Stellenbosch : Stellenbosch University, 2021-12) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
    ENGLISH ABSTRACT: The definable (p, q)-conjecture is a model-theoretic version of a (p, q)- theorem in combinatorics, which was expressed in the form of a question by A. Chernikov and P. Simon in 2015. Researchers have proved that the property holds for certain classes of structures. Based on those existing results, the main objective of the present thesis is to show that the definable (p, q)-conjecture holds for a dense pair of geometric distal structures that satisfies the following condition: algebraic closure and definable closure are the same in sense of the original geometric structure. Independently, we also explore a different approach to prove that under some conditions, the definable (p, q)-conjecture holds in certain cases for dense pairs of real closed ordered fields.
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    Imaginaries in dense pairs of real-closed fields
    (Stellenbosch : Stellenbosch University, 2017-03) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences
    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)