Stabilizing theorem in a category of almost completely decomposable Abelian groups
A class of almost completely decomposable Abelian groups is as an object of this article. A modern state of a structure theory of almost completely decomposable Abelian groups is described. Basic terms such as the controller of almost completely decomposable Abelian group, almost completely decomposable Abelian group with controller type (1,1,1), of index, which is a product of distinct primes. The purpose of the article is a construction investigation of a structure of almost completely decomposable Abelian groups with controller of type (1,1,1), of index, which is a product of distinct primes at growing of quantity of factors in product. In the main part an author constructed bijective correspondence between groups from the class of almost completely decomposable Abelian groups with a fixed partially ordered set of controller types and representations of partially ordered set of types of controller over the ring of residue classes modulo square-free, preserving isomorphisms, direct sums and indecomposable objects. The author set bijective correspondence between the indecomposable representations of finite representable partially ordered set of types of controller over the ring of residue classes modulo square-free and with basis of Hilbert of monoid solutions of a system of linear homogeneous Diophantine equations. The author investigated a monoids solutions structure of systems of Diophantine equations and multivariate Hilbert series associated with the category of almost completely decomposable Abelian groups with regulators, types of direct summands which are as partially ordered set (1,1,1). The author proved that components of a vector-dimension of indecomposable representations of partially ordered sets (1,1,1) over the ring of residue class modulo, which is a product of distinct primes, do not increase with an increase of quantity of factors in product. Results can be applied in a theory of Abelian groups and a theory of Artinian modules.