EZADS inputs which produce half-factorial block monoids.
Let \(A=\{a_0=1,a_1,\ldots,a_n\}\) be a set of pairwise co-prime integers, put \(q=\prod_{i=1}^na_i\), and let \(S=S(A):=\{q_0,q_1,\ldots,q_n\}\subset C_q\), where \(q_i\) is the residue class mod \(q\) containing \(q/a_i\). The block monoid \(B(C_q,S)\) associated with \(S\) is defined as the monoid consisting of all elements \(B=\prod_{s\in S}s^{v_s}\) (with \(v_s\) being non-negative integers), satisfying \(\sum_{s\in S}v_ss=0\). The aim of the author is to describe sets \(A\) for which \(B(C_q,S(A))\) is half-factorial, i.e., for each of its elements \(a\) all factorizations of \(a\) into irreducibles have the same length. He observes (Proposition 2.1) that from a result of \textit{S. T. Chapman} and \textit{W. W. Smith} [Isr. J. Math. 71, No. 1, 65-95 (1990; Zbl 0717.13014)] it follows that for \(n=2\) all monoids \(B(C_q,S)\) are half-factorial, and shows (Theorem 3.2) that if \(A_1\subset A_2\), and \(B(C_q,S(A_2))\) is half-factorial, then also \(B(C_q,S(A_1))\) is half-factorial. This leads to an inductive way of classifying all half-factorial monoids \(B(C_q,S)\).
scientific article; zbMATH DE number 6493124
scientific article