Equivalence of complete and partial automaton semigroups/monoids

Determine whether every (partial) automaton semigroup or automaton monoid—generated by a non-complete deterministic letter-to-letter transducer—can also be generated by some complete deterministic letter-to-letter transducer; equivalently, ascertain whether the class of complete automaton semigroups and monoids coincides with the class of partial automaton semigroups and monoids.

Background

The paper works with complete deterministic letter-to-letter transducers and their generated automaton semigroups/monoids, while noting that these notions extend to non-complete automata (yielding partial automaton semigroups/monoids). Whether every partial instance can be represented by a complete one is a foundational unresolved question about the scope of the model.

Resolving this would clarify the relationship between two widely used definitions in the literature and determine if results stated for complete automata automatically cover the partial setting.