Max-end invariance under \((\Phi,\Psi)\)-mutations of free commutative monoids. (Q744866)

In this article, the authors extend a result of a previous paper [\textit{K.-T. Howell, C. J. Maxson},``\((\phi,\psi)\)-mutations of commutative monoids with maximal endomorphism semirings'', Quaest. Math. 36, No. 4, 477-485 (2013)] and show that the max-end property is invariant under \((\Phi,\Psi)\)-mutations of the following classes of groups: free abelian groups, finitely generated abelian groups, bounded abelian groups, and the group direct sum \(\sum_{\lambda\in\Lambda}\mathbb Z_{p(\lambda)}\), where \(\Lambda\) is an indexing set and \(p\) is any function from \(\Lambda\) to the set of prime integers. It is also shown that the max-end property is invariant under \((\Phi,\Psi)\)-mutations of free commutative monoids with two generators, but stated that the problem is open for free commutative monoids with more than two generators.