2023/11/13 by Hannah Fox, Fox, Hannah, Agastya Goel +3
Mathematics · Computer Science · #Rings, Modules, and Algebras #semigroups and automata theory #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2311.07060
A subset S of an integral domain is called a semidomain if the pairs (S,+) and (S∖\0\, ⋅) are commutative and cancellative semigroups with identities. The multiplication of S extends to the group of differences \mathscrG(S), turning \mathscrG(S) into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e., semidomains S for which either s ∈ S or -s ∈ S for every s ∈ \mathscrG(S)). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the article, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.