Homogenizable structures and model completeness (Q334999)

A structure \(\mathcal{M}\) is called homogenizable if it is homogeneous after adding a finite number of new relations which are 0-definable. Here, the author divides the homogenizable structures into several classes and investigates the properties of the structures in these classes. He considers known examples with respect to his classes. A homogenizable structure \(\mathcal{M}\) is boundedly homogenizable if for each \(\bar{a} \in M\) there exists a \(\bar{b} \in M\) such that tp\((\bar{a}\bar{b})\) is isolated by a quantifier free formula, and \(\mathcal{M}\) is uniformly homogenizable if there is a tuple \(\bar{a}\in M\) such that for any \(\bar{b} \in M\), tp\((\bar{a}\bar{b})\) is isolated by a quantifier-free formula. The author shows that model completeness is important when studying homogenizability. He gives a necessary and sufficient condition for \(\aleph_0\)-categorical model-complete structures to be homogenizable. He shows: If \(\mathbf{K}\) is a class of structures closed under isomorphism and satisfying HP and AP, then the Fraïssé limit of \(\mathbf{K}\) is the unique boundedly homogeneous structure with age \(\mathbf{K}\).