Normal forms for connectedness in categories

The paper gives a simple result on the existence of normal forms for the following equivalence relation between objects of a category: A B if and only if there are maps A ?! B and B ?! A, under the hypothesis that the category has epi-mono factorizations and each object has nitely many sub-objects and quotient-objects. Applications to algebra, logic, automata theory, databases are presented. General abstract principles are useful in identifying patterns and in research. This paper presents one such principle, a simple rewriting result which holds in general categories. It is essentially the proof of facts like the existence of bases for nitely generated systems (vector spaces, systems of axioms, algebras), that a deterministic nite automata can be reduced to a minimal one, that some database queries can be reened and minimized, etc. It turns out that the scope of the principle is much wider than these examples. We present the background material and framework in Section 1. The statement and the proof of the principle is in Section 2. Then, Section 3 presents some well known cases where it is applied. Finally, Section 4 shows new results proved with the help of this principle. We will suppose a light knowledge of category theory as in the rst chapters of 10]. For rewriting theory we recommend 9] and the book 3]. 1.1. The equivalence relation. Let C be a category. We are interested in the following relation which occurs in some categories. Deenition 1. Let A; B be objects of C. Then A B if and only if there are arrows A ?! B and B ?! A. Using the properties of a category, it follows that is an equivalence relation. For example, in a partial order viewed as a category, the relation means equality of the elements of the partial order; in the category of sets