On the dualization of subdirect embeddings

Abstract

In the algebra there are several kinds of structure theorems which can be formulated without operations, using only homological tools. For instance, the wellknown fact that any universal algebra can be subdirectly embedded in a direct product of subdirectly irreducible algebras, can be formulated in a pure categorytheoretical manner. Now the question arises what its dual statement asserts. Our purpose is to give such a category which satisfies certain selfdual conditions, and making use of these, to prove structure theorems and their dual statements. The structure theorems themselves are, of course, well-known statements for algebraic stuctures. However, their duals yield some theorems of unusual type. About the possibility of the dualization there occurs some trouble. The most of the difficulties is at finding selfdual conditions being necessary to prove the theorems. So we must not make use of the condition 'every epimorphism is a normal one' which is fulfilled for groups, since its dual is false. Further the lattice of all congruencerelations of any universal algebra is a so-called compactly generated lattice. This fact plays a very important role in the proof of the theorem according to subdirect embeddings of universal algebras, nevertheless compactly generating is not a se!fdual notion. Applying the theorems proved for certain categories, we establish some particular theorems for rings, groups, modules, respectively. In w 2 we give a detailed enumeration of the usual notions and assertions of the theory of categories with respect to the importance of the dual notions and assertions, moreover, we form a system of selfdual conditions which will be satisfied by the category we are dealing with. w 3 is devoted to the investigation of subdirect embeddings, subdirect irreducibility and to the dualization of those. In w 4 we applicate the results developed before for rings, groups, module ~ and abelian groups. Most of the applications are concerned with rings.