Retractable state-finite automata without outputs

TL;DR

This paper provides a comprehensive characterization of finite-state retractable automata without outputs, focusing on the existence of specific homomorphisms onto their subautomata.

Contribution

It offers a complete description of the structure of state-finite retractable automata without outputs, extending previous work on special retractable automata.

Findings

Characterization of retractable automata without outputs

Conditions for the existence of retract homomorphisms

Structural properties of subautomata and automorphisms

Abstract

A homomorphism of an automaton ${\bf A}$ without outputs onto a subautomaton ${\bf B}$ of ${\bf A}$ is called a retract homomorphism if it leaves the elements of $B$ fixed. An automaton ${\bf A}$ is called a retractable automaton if, for every subautomaton ${\bf B}$ of ${\bf A}$, there is a retract homomorphism of ${\bf A}$ onto ${\bf B}$. In [1] and [3], special retractable automata are examined. The purpose of this paper is to give a complete description of state-finite retractable automata without outputs.