EZADS inputs which produce half-factorial block monoids

Abstract

We study the factorization properties of block monoids $$\fancyscript{B}(\mathbb {Z}/q\mathbb {Z},S)$$ resulting from the EZADS construction given in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012). This construction, which is inspired by a paper of Erdos and Zaks (J Number Theory 36 (1): 89–94, 1990), takes a finite set of integers (called an EZADS input) and produces a weakly half-factorial set called an EZADS. We are primarily interested in determining which EZADS inputs will produce a half-factorial EZADS. In Sect. 3 we show that if an EZADS input produces a half-factorial EZADS, then so will any of its subsets. In Sect. 4 we derive a bound which significantly simplifies the problem of determining whether an EZADS is half-factorial. In Sect. 5, we show how this bound may be used to reformulate the problem in terms of continuous quantities, then in Sect. 6 we apply these ideas to study four-element EZADSs. We describe a finite algorithm which, for fixed $$m$$ , can be used to classify all half-factorial EZADSs in the form $$\{\overline{1},\overline{ab},\overline{mb},\overline{ma}\}\subseteq \mathbb {Z}/mab\mathbb {Z}$$ . Finally in Sect. 7 we show that a special sequence of EZADSs studied in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012) are always half-factorial. We close by describing possible extensions of our work and related conjectures.