On minimum delta set values in block monoids over cyclic groups

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 where . We will show that the greatest integer r, denoted by , which divides every difference in lengths of factorizations in can be immediately determined by considering the continued fraction of . We then consider the set including necessary and sufficient conditions (which depend on p) for a value to be an element of . 2000 Mathematics Subject Classification Primary—20M14, 11A55, 20D60, 11A51 Parts of this work are contained in the first author’s Doctoral Dissertation written at the University of North Carolina at Chapel Hill under the direction of the third author.