Dependence logic in pregeometries and \(\omega\)-stable theories (Q2805022)

The authors present axiomatic descriptions of several forms of dependence and independence structures and introduce several forms of dependence (independence) logics with the so called independence (dependence) atoms of that or other kind as the only formulas. They show that the completeness question for these logics admits a positive answer with respect to naturally arising classes of dependence (independence) structures in the contexts of databases, pregeometries (matroids) and \(\omega\)-stable first-order theories, in particular, theories of infinite vector spaces over a countable field and algebraically closed fields of fixed characteristic. The authors conclude that ``uses of independence concepts in as different areas as database theory, algebra and model theory can be completely characterized by the same axioms''.