Compatible Pairs of Low-Dimensional Associative Algebras and Their Invariants

A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of complex dimension less than four. Alongside these classifications, we systematically compute and analyze various algebraic invariants associated with them, including derivations, centroids, automorphism groups, quasi-centroids, Rota-Baxter operators, Nijenhuis operators, averaging operators, Reynolds operators, quasi-derivations, and generalized derivations.