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

Abstract

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.

Minimal presentations for monoids with the ascending chain condition on principal ideals

“…The idea underneath the computation of the catenary degree in these papers is that c(S) = sup b∈Betti(S) c(b) (see [5,Corollary 8]; its proof is based on [5,Lemma 7] which has some mistakes, corrected later by the same author in [6] by adding the ascending chain condition). In [2,Theorem 2.5] it is also shown that the maximum of Δ(S) is reached in the Betti elements of S. Hence, we get the following consequence. …”

“…Along this line, characterizations of the catenary degree were first performed for finitely generated monoids (see for instance [1]), and then extended to a non-finitely generated setting in [5], showing in both cases that the catenary degree is reached in the Betti elements of the monoid. Then in [2] it was proved that the same occurs with the maximum of the Delta sets of a BF-monoid (BF-monoid means that for a fixed element the lengths of its factorizations is bounded, and these monoids might not be finitely generated; see [4] for more details).…”

numericalsgps, a GAP package for numerical semigroups

“…. , l t − l t−1 } is the Delta set associated to s. The Delta set of S is the union of all the Delta sets of s. This set is finite, and its maximum is achieved in one of the Betti elements of S ( [17]). …”

Factorization length distribution for affine semigroups I: Numerical semigroups with three generators

“…. , n 2 − n 1 ), which arises in the semigroup literature as the minimum element of the delta set (see [19] for more on this invariant). Since n i ≡ n j (mod δ) for every i, j, it follows that δ is the smallest distance that can occur between distinct factorization lengths of n, meaning all factorization lengths of a given n are equivalent modulo δ.…”