Divisibility Monoids: Presentation, Word Problem, and Rational Languages

We present three results on divisibility monoids. These divisibility monoids were introduced in [11] as an algebraic generalization of Mazurkiewicz trace monoids. (1) We give a decidable class of presentations that gives rise precisely to all divisibility monoids. (2) We show that any divisibility monoid is an automatic monoid [5]. This implies that its word problem is solvable in quadratic time. (3) We investigate when a divisibility monoid satisfies Kleene’s Theorem. It turns out that this is the case iff the divisibility monoid is a rational monoid [25] iff it is width-bounded. The two latter results rest on a normal form for the elements of a divisibility monoid that generalizes the Foata normal form known from the theory of Mazurkiewicz traces.