On Direct Product and Quotient of Strongly Connected Automata

TL;DR

The paper addresses the structure of the direct product $A × B$ of a strongly connected permutation automaton $A$ and a strongly connected synchronizing automaton $B$. It develops two automaton congruences, $π$ and $ρ$, from the minimal transition ideal $I_A$ and its idempotents, proving that $A × B$ is a quasi-ideal automaton with $I_{A×B}$ a right group and the idempotent ranges partitioning the state set. It then shows $A × B$ is isomorphic to the direct product of its quotients $(A × B)/π$ and $(A × B)/ρ$, thereby recovering $A$ and $B$ as these quotients: $A ≅ (A × B)/π$ and $B ≅ (A × B)/ρ$. This yields a canonical, constructive decomposition for the direct product and generalizes prior results on state-independent decompositions to the broader case of non-state-independent permutation automata.

Abstract

Let $A \ \times \ B$ be the direct product of a strongly connected permutation automaton $A$ and a strongly connected synchronizing (reset) automaton $B$, then $A \ \times \ B$ is strongly connected and $$\boldsymbol{A \cong (A \times B)/π}$$ $$\boldsymbol{B \cong (A \times B)/ρ}$$ $$\boldsymbol{(A \times B) \ \cong \ (A \times B)/π\ \times \ (A \times B)/ρ}$$ where $π$ and $ρ$ are automaton congruence relations defined in this paper, $(A \times B)/π$ and $(A \times B)/ρ$ are quotient automata constructed by $π$ and $ρ$ respectively.

Theorems & Definitions (26)

• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof