2 research outputs found
Primitive Automata that are Synchronizing
A deterministic finite (semi)automaton is primitive if its transition monoid
(semigroup) acting on the set of states has no non-trivial congruences. It is
synchronizing if it contains a constant map (transformation). In analogy to
synchronizing groups, we study the possibility of characterizing automata that
are synchronizing if primitive. We prove that the implication holds for several
classes of automata. In particular, we show it for automata whose every letter
induce either a permutation or a semiconstant transformation (an idempotent
with one point of contraction) unless all letters are of the first type. We
propose and discuss two conjectures about possible more general
characterizations.Comment: Note: The weak variant of our conjecture in a stronger form has been
recently solved by Mikhail Volkov arXiv:2306.13317, together with several new
results concerning our proble