Khripchenko N. Finitary incidence rings

In the thesis we introduce the notion of a partially ordered category (a pocategory for short) and its finitary and weak incidence rings. We describe the invertible elements, the Jacobson radical, the idempotents and the regular elements of the finitary incidence ring. To each quasiordered set (qoset for short) and ring we associate a pocategory. The finitary incidence ring of this pocategory is a generalization of the notion of the incidence ring of a locally finite qoset. For the finitary and weak incidence rings of the pocategories associated with the qosets and indecomposable rings a positive solution of the isomorphism problem and an isomorphism of the groups of outer automorphisms of these rings with the group of outer automorphisms of the pocategory are obtained. The group of outer automorphisms of the associated pocategory is fully described if the classes of the qoset are finite and the ring is local. We prove that the Lie rings of outer derivations of the finitary and weak incidence rings are isomorphic to the Lie ring of outer derivations of the pocategory. For the pocategories, associated with a qoset and a ring, this Lie ring is described.