The term "Associative bialgebroid" does not appear to be a widely recognized or established concept in the mathematical literature, particularly within the fields of algebra, category theory, or Hopf algebra theory where related terms such as "bialgebra" and "bialgebroid" are standard.
A "bialgebroid" is a generalization of a bialgebra structure over noncommutative base algebras, typically involving a total algebra and a base algebra with compatible coalgebra-like and algebraic structures. The prefix "associative" usually refers to algebraic structures in which the binary operation satisfies the associative property. However, in standard definitions of bialgebroids, the total algebra is already assumed to be associative, making the qualifier "associative" redundant unless used for emphasis.
No authoritative or peer-reviewed sources explicitly define or develop a concept named "associative bialgebroid" as a distinct or specialized structure. As such, accurate information is not confirmed regarding its definition, properties, or applications.
It is possible that the term arises in specialized or emerging research contexts, or as an informal descriptor in works not widely disseminated. Without verifiable sources, the term remains outside established mathematical terminology.