Characterizations of normal cancellative monoids

<abstract><p>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.</p></abstract>