On the delta set and the Betti elements of a BF-monoid

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of �( S) and the Betti elements of S. We prove how the minimum and maximum element of �( S) can be determined using the Betti elements of S. This leads to a determination of when �( S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.