chapter  19
11 Pages

A Note on Sets of Lengths of Powers of Elements of Finitely Generated Monoids

ByWolfgang Hassler

By a monoid we always mean a (usually multiplicatively written) commutative semigroup H with identity element for which the cancellation law holds (i.e. ab = ac implies b = c for all a, b, c ∈ H). Let H be a monoid. An element u ∈ H\H× (where H× denotes the group of invertible elements of H) is called irreducible (or an atom) if u = ab implies that either a ∈ H× or b ∈ H× for all a, b ∈ H . The monoid H is called atomic if every nonunit a ∈ H possesses a factorization

a = u1 · . . . · un (19.2) into irreducible elements ui of H . The integer n is called the length of the factorization (19.2). We call

L(a) = LH(a) = {n ∈ N | n is length of some factorization of a} the set of lengths of a. During the last years, much effort was made to determine the structure of sets of lengths for certain classes of domains and monoids, e.g. finitely generated monoids, orders of global fields, Krull monoids, Congruence monoids, one-dimensional local domains and higher dimensional algebras over perfect fields. The reader is referred to [1], [5], [6], [12], [14] and [13] for the most recent results in this area. It turned out that sets of lengths in all mentioned classes of rings and monoids (in some cases under additional finiteness assumptions) have the following special structure: they are, up to bounded initial and final segments, a union of arithmetical progressions with bounded distance (see Definition 1.2 and Theorem 1.3 for the detailed statement in the case of finitely generated monoids). As a consequence of this structure theorem, one can classify sets of lengths by their initial and final segment and the period of their central part. Obviously, there are only finitely many classes (called “types”, see Definition 2.1). Very little is known about the behavior of the type of sets of lengths, yet. For example, given two elements a and b, what can be said about the type of L(ab) if the type of L(a) and L(b) is known? What kind of additional information on a and b is needed to determine L(ab)? In view of efficient methods for the computation of sets of lengths (and for theoretical reasons) it would be desirable to have “simple” criteria to determine the type of the sets of lengths of (at least certain classes) of elements. In this note we provide such a criterion for powers of elements of a finitely generated monoid H . Our research was initiated by the following question raised by F. Halter-Koch: let a ∈ H\H× (assuming that H is finitely generated) and consider the sequence L(an) of sets of lengths. Do the initial and final segments of L(an) repeat periodically if n grows?