Kinds of pregeometries of acyclic theories

The article is devoted to the description of types of pregeometries with an algebraic closure operator for acyclic theories. In these theories we describe conditions of violation of the exchange property for a pregeometry. Taking into account these conditions, we introduce new concepts that do not rely on the exchange property: $a$-pregeometry, $a$-modularity, etc. The dependence conditions for an $a$-modular and $a$-locally finite $a$-pregeometry on the number of non-isomorphic trees and special points are established. Sufficient conditions of dependence for a $a$-local finite $a$-pregeometry on the vertices of the $a$-type are established, too.