On minimum delta set values in block monoids over cyclic groups

Abstract

The difference in length between two distinct factorizations of an element in a Dedekind domain or in the corresponding block monoid is an object of study in the theory of non-unique factorizations. It provides an alternate way, distinct from what the elasticity provides, of measuring the degree of non-uniqueness of factorizations. In this paper, we discuss the difference in consecutive lengths of irreducible factorizations in block monoids of the form $$\mathcal{B}_a(n)=\mathcal{B}(\mathbb{Z}_n, S)$$ where $$S = \{ 1+ n \mathbb{Z}, a+n \mathbb{Z} \}$$ . We will show that the greatest integer r, denoted by $$\delta_2(a,n)$$ , which divides every difference in lengths of factorizations in $$\mathcal{B}_a(n)$$ can be immediately determined by considering the continued fraction of $$\frac{n}{a}$$ . We then consider the set $$\delta_2(p)=\{\delta_2(a,p)\mid 1<a<p\}$$ for a prime p, which has been shown to be a subset of [1, p−2]. Various results are established regarding the structure of $$\delta_2(p)$$ including necessary and sufficient conditions (which depend on p) for a value $$d > \sqrt{p}$$ to be an element of $$\delta_2(p)$$ .