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Completely Reachable Almost Group Automata

David Fernando Casas Torres

Abstract

We consider finite deterministic automata such that their alphabets consist of exactly one letter of defect 1 and a set of permutations of the state set. We study under which conditions such an automaton is completely reachable. We focus our attention on the case when the set of permutations generates a transitive imprimitive group.

Completely Reachable Almost Group Automata

Abstract

We consider finite deterministic automata such that their alphabets consist of exactly one letter of defect 1 and a set of permutations of the state set. We study under which conditions such an automaton is completely reachable. We focus our attention on the case when the set of permutations generates a transitive imprimitive group.
Paper Structure (8 sections, 14 theorems, 20 equations, 2 figures)

This paper contains 8 sections, 14 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A necessary condition
  4. Rystsov graphs of almost group automata
  5. The structure of Rystsov graphs
  6. Intermezzo
  7. Non-reachability and invariance
  8. Conclusion

Key Result

Theorem 1

Let $G$ be a permutation group on $[n]$ with $n \geq 3$. Then $G$ is primitive if and only if for each [idempotent] transformation $f: [n] \rightarrow [n]$ of defect 1 every non-empty subset $A \subseteq [n]$ is reachable in $\langle G \cup \{f\} \rangle$.

Figures (2)

  • Figure 1: The initial edges of $\Gamma_{1}(\mathcal{E}_{18})$.
  • Figure 2: A strongly connected component of $\Gamma_{1}(\mathcal{E}_{18})$.

Theorems & Definitions (23)

  • Theorem 1: hoffman2023primitive[Theorem 3.1]
  • Proposition 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Lemma 3
  • ...and 13 more