In this paper, we present three limit representations of the core-EP inverse. The first approach is based on the full-rank decomposition of a given matrix. The second and third approaches, which depend on the explicit expression of the core-EP inverse, are established. The corresponding limit representations of the dual core-EP inverse are also given. In particular, limit representations of the core and dual core inverse are derived.

This paper is devoted to constructing quasitriangular bialgebras (or Hopf algebras). The tool we use is a new coproduct, we call it the unified coproduct, in the construction of which a Hopf algebra and an algebra are connected by three algebra maps: two coactions and a generalized cocycle. Both the crossed coproduct of a bialgebra coacting on a coalgebra and the bicrossed coproduct of two bialgebras are special cases of the unified coproduct. Then we establish a bijective correspondence between the set of all quasitriangular structures on the arbitrary unified coproduct and a certain set of datum related to the components of the unified coproduct. As the main application, we derive the necessary and sufficient conditions for the crossed coproduct (or bicrossed coproduct) to be a quasitriangular Hopf algebra.