Finite transducers for divisibility monoids

Divisibility monoids are a natural lattice-theoretical generalization of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the involved divisibility lattices is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be commutations. Here, we show that every divisibility monoid admits an explicit finite transducer which allows to compute normal forms in quadratic time. In addition, we prove that every divisibility monoid is biautomatic.