The set of minimal distances in Krull monoids (Q2809363)

For a non-unit \(a\) of a Krull monoid \(H\), let \(L(a)=\{m_1<m_2<\cdots<m_k\}\) be the set of lenghts of distinct factorizations of \(a\) into atoms, and denote by \(\Delta(L(a))\) the set \(\{m_i-m_{i-1}:\;i=2,\dots,k\}\). The set of distances of \(H\) is defined by \(\Delta(H)=\bigcup_{a\in H}\Delta(L(a)\). The authors study the maximal elements of the set of minimal distances \(\Delta^*(H)\) of \(H\) defined by \(\Delta^*(H)=\min_S\Delta(S)\), where \(S\) ranges over all divisor-closed submonoids of \(H\) satisfying \(\Delta(S)\neq0\). Their main result (Theorem 1.1) states that if the class-group \(G\) of \(H\) satisfies \(3\leq|G|<\infty\), then NEWLINE\[NEWLINE\max\Delta^*(H)\leq\max\{\exp(G)-2,r(G)-1\}NEWLINE\]NEWLINE with equality in the case when every class contains a prime divisor. Here, \(\exp(G)\) denotes the exponent of \(G\) and \(r(G)\) is the rank of \(G\). This result is supplemented by a converse theorem (Theorem 4.5) dealing with the structure of minimal non-half-factorial subsets \(G_0\) of \(G\) satisfying \(\min\Delta(B(G_0))=\max\Delta^*(B(G))\), \(B(G),B(G_0)\) being the monoids of zero-sum sequences over \(G\), resp. \(G_0\).