Kinds of Pregeometries of Cubic Theories

Abstract

The description of the types of geometries is one of the main problems in the structural classification of algebraic systems. In addition to the well-known classical geometries, a deep study of the main types of pregeometries and geometries was carried out for classes of strongly minimal and $\omega$-stable structures. These studies include, first of all, the works of B.I.~Zilber and G.~Cherlin, L.~Harrington, A.~Lachlan in the 1980s. Early 1980s B.I.~Zilber formulated the well-known conjecture that the pregeometries of strongly minimal theories are exhausted by the cases of trivial, affine, and projective pregeometries. This hypothesis was refuted by E.~Hrushovski, who proposed an original construction of a strongly minimal structure that is not locally modular and for which it is impossible to interpret an infinite group. The study of types of pregeometries continues to attract the attention of specialists in modern model theory, including the description of the geometries of various natural objects, in particular, Vamos matroids. In this paper we consider pregeometries for cubic theories with algebraic closure operator. And we notice that for pregeometries $\langle S,{\rm acl}\rangle$ in cubic theories, the substitution property holds if and only if the models of the theory do not contain infinite cubes, in particular, when there are no finite cubes of unlimited dimension. By virtue of this remark, we introduce new concepts of $c$-dimension, $c$-pregeometry, $c$-triviality, $c$-modularity, $c$-projectivity and $c$-locally finiteness. And besides, we prove the dichotomy theorem for the types of $c$-pregeometries.

Keywords

• pregeometry
• cubic theory
• $c$-pregeometry
• $c$-triviality
• $c$-modularity
• $c$-projectivity
• $c$-locally finiteness