  An integral domain $R$ is atomic if each nonzero nonunit of $R$ factors into
irreducibles. In addition, an integral domain $R$ satisfies the ascending chain
condition on principal ideals (ACCP) if every increasing sequence of principal
ideals (under inclusion) becomes constant from one point on. Although it is not
hard to verify that every integral domain satisfying ACCP is atomic, examples
of atomic domains that do not satisfy ACCP are notoriously hard to construct.
The first of such examples was constructed by A. Grams back in 1974. In this
paper we delve into the class of atomic domains that do not satisfy ACCP. To
better understand this class, we introduce the notion of weak-ACCP domains,
which generalizes that of integral domains satisfying ACCP. Strongly atomic
domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in
1990. It turns out that every weak-ACCP domain is strongly atomic, and so we
introduce a taxonomic classification on our class of interest: ACCP implies
weak-ACCP, which implies strong atomicity, which implies atomicity. We study
this chain of implications, putting special emphasis on the weak-ACCP property.
This allows us to provide new examples of atomic domains that do not satisfy
ACCP.
