Morfismidest kokorrutiste vahel järjestatud ja additiivsel juhul

Abstract

In mathematics morphisms between various structures are often studied. In particular, the endomorphisms of a structure can be of interest. The collection of endomorphisms of a given structure is a monoid. Studing that monoid can be easier, if the monoid is decomposed into simpler monoids using some construction. One such construction was given by Vladimir Flaicher and Ulrich Knauer in 1988. They proved that the endomorphism monoid of an act over a monoid is isomorphic to the wreath product of some monoid and some small category. In the present work we try to generalize that result. We interpret the essence of the theorem to be, that to know morphisms between objects and the composition of these morphisms, it suffices to know the morphisms between suitable subobjects of the given objects and the compositions of these morphisms between subobjects. The theorem we mentioned gives a result of that sort for the representation of the endomorphism monoid of an act over a monoid. We generalize the result in different directions. For one, we do not restrict ourselves to any specific category, but try to give a result for all categories. Another direction in which we generalize the result, is that we try to represent full subcategories of given categories not only endomorphism monoids. The third direction of generalization is enrichment. To be more precise, we look at two specific cases of enrichment. We hope that in doing so, finding a common generalization for them might become easier. We view enrichment over the category of partially ordered sets and enrichment over the category of Abelian groups. The case of partially ordered sets is similar to the case of ordinary categories. Indeed, ordinary categories can be viewed as order enriched categories with discrete order. We prove a generalization of [8] theorem II.7.7 and see how what we proved relates to it. We also prove a result relating to the existence of nice decompositions for a certain subclass of categories. The case of enrichement over Abelian groups is somewhat different from the case of ordinary categories. It does not generalize it. We prove an analogue of [8] theorem II.7.7. And present some simple, known results, that the author found here and there, that clarify the situation and help in applying the proven theorem. We also present, without proof, a couple of known results in the case of modules over a ring.