A monoid is hereditarily atomic exactly when all its undermonoids are atomic, this forces the ACCP, and torsion abelian groups are precisely the monoids all of whose undermonoids are half- or length-factorial.
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Arithmetic properties encoded in undermonoids
A monoid is hereditarily atomic exactly when all its undermonoids are atomic, this forces the ACCP, and torsion abelian groups are precisely the monoids all of whose undermonoids are half- or length-factorial.