### Divisibility Monoids: Presentation, Word Problem, and Rational Languages [chapter]

Dietrich Kuske
2001 Lecture Notes in Computer Science
We present three results on divisibility monoids. These divisibility monoids were introduced in  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  . 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
more » ... the case iff the divisibility monoid is a rational monoid  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. An algebraic characterization of trace monoids was given only later by Duboc . Divisibility monoids are a lattice theoretically easy generalization of these algebraic conditions. Our first result (Theorem 4) describes a decidable class of presentations that give rise precisely to all divisibility monoids. Since the canonical presentations for trace monoids belong to this class, our result can be seen as an extension of Duboc's characterization to the realm of divisibility monoids. For trace monoids, the word problem can be solved in linear time  . From our presentation result, an exponential algorithm for the word problem in divisibility monoids follows immediately. But we show that one can do much better: The work on automatic groups  has been generalized to the realm of semigroups. Intuitively, a semigroup is automatic if it admits a presentation such that the equality can be decided by an automaton and such that the multiplication by generators can be performed by an automaton [17, 5] . In particular, Campbell et al.  showed that the word problem for any automatic semigroup is solvable in quadratic time. Theorem 8 shows that any divisibility monoid is an automatic semigroup. Hence, we can infer from the result of Campbell et al. that the word problem for any divisibility monoid can be solved in quadratic time. We do not know whether this result can be improved, but we have serious doubts that a linear time algorithm exists. Kleene  showed that in a free finitely generated monoid the recognizable languages are precisely the rational ones. It is known that in general this is false, but Kleene's result was generalized in several directions, e.g. to formal power series by Schützenberger  , to infinite words by Büchi , and to rational monoids by Sakarovitch  . In all these cases, the notions of recognizability and of rationality were shown to coincide. This is not the case in trace monoids any more. Even worse, in any trace monoid (which is not a free monoid), there exist rational languages that are not recognizable. But a precise description of the recognizable languages in trace monoids using c-rational expressions could be given by Ochmański  . A further generalization of Kleene's and Ochmański's results to concurrency monoids was given in  . The proofs by Ochmański as well as by Droste heavily used the internal structure of the elements of the corresponding monoid. The original motivation for the consideration of divisibility monoids in  was the search for an algebraic version of these proofs. We succeeded showing that in a divisibility monoid with finitely many residuum functions, the recognizable languages coincide with the (m)c-rational ones (cf.  for precise definitions of these terms). Thus, two main directions of generalization of Kleene's Theorem in monoids are represented by Sakarovitch's rational monoids and by trace monoids. Since the only trace monoids that satisfy Kleene's Theorem are free monoids, these two directions are "orthogonal", i.e. the intersection of the classes of monoids in consideration is the set of free monoids. In  we already remarked that there are divisibility monoids that satisfy Kleene's Theorem and are not free. Thus, our further extension of Ochmański's result to divisibility monoids  is not "orthogonal" any more. In this paper, we describe the class of divisibility monoids that satisfy Kleene's Theorem. Essentially, Theorem 13 says that a divisibility monoid satisfies Kleene's Theorem if and only if it is rational if and only if it is width-bounded. Thus, in the context of divisibility monoids,