Wild and Wooley Numbers

That is, W0 is the set of all finite products of the generators, allowing repetitions. Let W signify the larger multiplicative semigroup generated by W0 together with {1/2}. We call W the wild semigroup and W0 the Wooley semigroup. The question we consider is: Which integers belong to these semigroups? The sets of integer elements W(Z) := W ∩ Z and W0(Z) := W0 ∩ Z themselves form multiplicative semigroups, which we term the wild integer semigroup and Wooley integer semigroup, and we refer to their members as “wild integers” and “Wooley integers,” respectively. We have the immediate implication that each Wooley integer is a wild integer, but the converse need not hold. The Wooley semigroup W0 is a semigroup without unit, whereas the wild semigroup W is a semigroup with unit, and the semigroups W0(Z) and W(Z) inherit these properties. Our particular choice of terminology is explained at the end of the introduction. An irreducible element of a commutative semigroup T is one that cannot be written as a product of two nonunits (noninvertible elements) in the semigroup (see Gilmer [8, sec. 1.6]). We call the irreducible elements of the wild integer semigroup wild numbers; similarly we christen the irreducible elements of the Wooley integer semigroup Wooley numbers. Thus the wild numbers are a subset of the wild integers, and the Wooley numbers are a subset of the Wooley integers. It is immediately evident that 2 is both a wild number and a Wooley number. It is also easy to show that 3 is not a wild number, hence not a Wooley number. However the nature of other wild numbers or Wooley numbers is less apparent. The object of this paper is to determine properties of wild numbers and Wooley numbers. It turns out that the Wooley numbers have a complicated and not completely understood structure; by comparison, the wild numbers have a reasonably simple description. The wild numbers and the Wooley numbers differ in some significant ways. An odd integer w is in the wild integer semigroup if and only if there is a nonnegative integer j such that 2 jw is in the Wooley integer semigroup. At the level of irreducible elements, we infer that if w is a wild number, then 2 jw is a Wooley number for some j ; if 2 jw is a Wooley number, however, we cannot (currently) decide whether w must be a wild number. At first glance the Wooley numbers seem to be the simpler objects from a computational perspective. In section 2 we show that there is an effectively computable procedure for deciding whether a given rational number r belongs to W0.