Omega-categoricity, relative categoricity and coordinatisation

Abstract

L and L − are countable first-order languages, L − ⊆L, and one of the symbols which is in L but not in L − is a 1-ary relation symbol P. The theory T is a complete theory in L with infinite models. We assume that if B is any model of T, the set P B of elements of B which satisfy P(x) is the domain of a substructure of the reduct B|L − of B to L − ; we write P(B) for this substructure. The paper will discuss two main questions about relative categoricity. Suppose T is relatively categorical and B is a model of T. Then we ask when the following hold: (1) B is explicitly definable in terms of P(B). (2) Every element of B is algebraic over P(B)