Arithmetic properties encoded in undermonoids

Abstract

Let M be a cancellative and commutative monoid. A submonoid N of M is called an undermonoid if the Grothendieck groups of M and N coincide. For a given property p , we are interested in providing an answer to the following main question: does it suffice to check that all undermonoids of M satisfy p to conclude that all submonoids of M satisfy p ? In this paper, we give a positive answer to this question for the property of being atomic, and then we prove that if M is hereditarily atomic (i.e., every submonoid of M is atomic), then M must satisfy the ACCP, proving a recent conjecture posed by Vulakh and the first author. We also give positive answers to our main question for the following well-studied factorization properties: the bounded factorization property, half-factoriality, and length-factoriality. Finally, we determine all the monoids whose submonoids/undermonoids are half-factorial/length-factorial.