The Structure of an Automaton and Its Operation-Preserving Transformation Group

TLDR: The main result shows the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.

Abstract: This paper is mot iva ted by Fleck's s tudy [1] on certain classes of structurepreserving, nontrivial t ransformations of au tomata . In tha t paper the class of those transformations which preserve \"strongly-connectedness\" is completely characterized. An interesting subclass, the class of operation-preserving functions (which are essentially homomorphisms) is introduced there. Fleck showed tha t the set of all operation-preserving functions of an au tomaton A onto itself constitutes a group G(A). In [2] some of the properties of G(A) when A is strongly connected were studied. I t was shown in the lat ter paper tha t corresponding to every finite group G of regular permutat ions there is a strongly connected automaton A for which G = G(A). Since, in fact, the group G(A) determines the structure of A, it would appear tha t the structure of G(A) and of A should be related. The present paper investigates tha t relationship. The main result shows tha t the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.