On Direct Product and Quotient of Strongly Connected Automata

TL;DR

This paper investigates the properties of the direct product of strongly connected automata, establishing conditions under which the product automaton remains strongly connected and characterizing its quotients.

Contribution

It introduces specific congruence relations for automata and demonstrates how automorphisms relate to quotient automata in the context of strongly connected automata.

Findings

The direct product of a strongly connected permutation automaton and a strongly connected synchronizing automaton is strongly connected.

Automorphisms of the automaton can be represented as quotients of the direct product.

Explicit congruence relations are constructed to recover original automata from the product.

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 $$\boldsymbol{A \cong (A \times B)/\pi}$$ $$\boldsymbol{B \cong (A \times B)/\rho}$$ $$\boldsymbol{(A \times B) \ \cong \ (A \times B)/\pi \ \times \ (A \times B)/\rho}$$ where $\pi$ and $\rho$ are automaton congruence relations defined in this paper, $(A \times B)/\pi$ and $(A \times B)/\rho$ are quotient automata constructed by $\pi$ and $\rho$ respectively.