Characterizations of normal cancellative monoids

Normal cancellative monoids were introduced to explore the general structure of cancellative monoids, which are innovative and open up new possibilities. Specifically, we pointed out that the Green's relations in a cancellative monoid $S$ are determined by its unitary subgroup $U$ to a great extent. The specific composition of egg boxes in $S$, derived from the general semigroup theory, can be settled by the subgroups of $U$. We call a cancellative monoid normal when these subgroups are normal and characterize it as an NCM-system. This NCM-system was created in this article and can be obtained by combining a group and a condensed cancellative monoid. Furthermore, we introduced the concept of torsion extension and proved that a special kind of normal cancellative monoids can be constructed delicately by the outer automorphism groups of given groups and some simplified cancellative monoids.