Covering Morphisms and Unique Minimal D-Schemes

  R. Tindell

 In this paper we answer the following question: given a D-scheme , is there a (unique) smallest D-scheme within the strong equivalence class of . In addition to providing an affirmative answer to this question, we provide a description of a reduction process leading from a D-scheme toits minimazition over the class of D- schemes.

 Since we are interested in D-schemes in this paper, we restrict our discussion to what, in the terminology of Elgot [CE], could be referred to as biscalar schemes whose outdegrees are bounded by 2. The paper is organized as follows. Section 2 gives the basic definitions of digraphs, schemes, and (homo)morphisms for these two classes and introduces the class of D-schemes. Section 3 recalls the definitions necessary to state the geometric characterization of D-Schemes from [BT] and introduces the notion of strong behavior and strong equivalence for schemes. Section 4 develops the basic properties of morphisms between schemes. Section 5 discusses the minimization process over the class of all schemes and states and proves the main theorem of the paper, Theorem 5.1. Section 6 is the final section and details the consequences of the main theorem.

 Gyenizse Pal 1996. Szeptember 4. Szerda 14:21:05 MET DST