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 Δ(S) and the Betti elements of S. We prove how the minimum and maximum element of Δ(S) can be determined using the Betti elements of S. This leads to a determination of when Δ(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.