Arithmetic of semisubtractive semidomains.

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 풢(S), turning 풢(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 ∈풢(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 paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains. [ABSTRACT FROM AUTHOR]