Erdős-Zaks all divisor sets.
Let \(G\) be an Abelian group, and \(S\) be a subset of \(G\). Let \(\mathfrak F(G,S)\) be the free Abelian group with basis \(S\), and \(\sigma\colon\mathfrak F(G,S)\to G\) be the monoid homomorphism mapping each element of \(S\) to itself. The block monoid \(\mathfrak B(G,S)\) is defined as \(\ker\sigma\). The authors study factorization properties of the monoid \(\mathfrak B(G,S)\), where \(G\) is a finite cyclic group. An all divisor set in \(\mathbb Z_n\) is a subset of \(\mathbb Z_n\) containing \(1\) such that the least residues of the elements in \(S\) are divisors of \(n\) in \(\mathbb Z\). Using Erdős-Zaks' work, the authors develop an algorithm that starts with an `EZADS' input (see title), and leads to a list of blocks which include all the non-primary irreducible blocks in \(\mathfrak B(\mathbb Z_q,S)\): here \(q\) is a positive integer, and \(S\) is an all divisor set (called an EZADS), both of them obtained from the EZADS input. The authors characterize the atoms in the list of blocks. They prove that the block monoid \(\mathfrak B(\mathbb Z_q,S)\) is weakly half-factorial, and obtain a criterion for half-factoriality of \(\mathfrak B(\mathbb Z_q,S)\) using the above mentioned list of blocks. Further results and numerical examples are included.
scientific article; zbMATH DE number 6083229
scientific article