On axiomatizability and decidability of universal theories of hereditary classes of matroids

Abstract

Two hereditary classes of matroids – the class of matroids of rank doesn’t exceeding a fixed positive integer k and the class of matroids of finite rank – are studied by means of the model theory. The problems of axiomatizability of these two classes of matroids as structures and the problems of algorithmic decidability of universal theories of these classes are considered. It is shown that the first class is finitely axiomatizable whereas the second one is nonaxiomatizable. Decidability of the universal theories of the both classes is proved.