Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

In this survey, we report our recent work concerning combination results for interpolation and uniform in the context of quantifier-free fragments first-order theories. We stress model-theoretic algebraic aspects connecting topic with amalgamation, strong model-completeness. give sufficient (and, relevant situations, also necessary) conditions transfer property to combined theories; investigate non-disjoint signature case under assumption that shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, provide a modular result existence interpolants. Model completions play key role whole paper: They enter into represent semantic counterpart