  A compatible associative algebra is a vector space equipped with two
associative multiplication structures that interact in a certain natural way.
This article presents the classification of these algebras with dimension less
than four, as well as the classifications of their corresponding derivations,
centroids, automorphisms, and quasi-centroids. We then characterize a selection
of further invariants such as Rota-Baxter operators and second cohomology for
some specific examples.
