Regular commutative semigroups

24 Sep 2023

Regular commutative semigroups (RCSs) are a generalization of abelian groups, in which the identity and inverse laws are weakened. They turn out to have an interesting internal structure; essentially, an RCS can be viewed as a “bundle” of abelian groups over a semilattice. In this paper, I prove many theorems about RCSs. I also define the concept of a “regular semiring,” which is like a ring, but has an RCS instead of an abelian group as its additive structure.