caa a22 4500 463177851 CHVBK 20180406164830.0 cr unu---uuuuu 170326e20070801xx s 000 0 eng 10.1007/s11139-006-9000-x doi (NATIONALLICENCE)springer-10.1007/s11139-006-9000-x On minimum delta set values in block monoids over cyclic groups [Elektronische Daten] [Sooah Chang, Scott Chapman, William Smith] 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 $$\delta_2(p)$$ . Springer Science + Business Media, LLC, 2006 Block monoid nationallicence Elasticity of factorization nationallicence Non-unique factorization nationallicence Continued fraction nationallicence Minimal zero-sequence nationallicence Chang Sooah Department of Mathematics, The University of North Carolina at Chapel Hill, 27599-3250, Phillips Hall, Chapel Hill, North Carolina, U.S.A aut Chapman Scott Department of Mathematics, Trinity University, One Trinity Place, 78212-7200, San Antonio, Texas, U.S.A aut Smith William Department of Mathematics, The University of North Carolina at Chapel Hill, 27599-3250, Phillips Hall, Chapel Hill, North Carolina, U.S.A aut The Ramanujan Journal Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 14/1(2007-08-01), 155-171 1382-4090 14:1<155 2007 14 11139 https://doi.org/10.1007/s11139-006-9000-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-006-9000-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Chang Sooah Department of Mathematics, The University of North Carolina at Chapel Hill, 27599-3250, Phillips Hall, Chapel Hill, North Carolina, U.S.A aut NATIONALLICENCE 700 1- Chapman Scott Department of Mathematics, Trinity University, One Trinity Place, 78212-7200, San Antonio, Texas, U.S.A aut NATIONALLICENCE 700 1- Smith William Department of Mathematics, The University of North Carolina at Chapel Hill, 27599-3250, Phillips Hall, Chapel Hill, North Carolina, U.S.A aut NATIONALLICENCE 773 0- The Ramanujan Journal Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 14/1(2007-08-01), 155-171 1382-4090 14:1<155 2007 14 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer