Cleft Coextension for symmetric twisted partial coactions on coalgebras

In this paper, we will introduce the concepts of symmetric twisted partial Hopf coactions, and discuss under which conditions a given symmetric twisted partial Hopf coaction is globalizable. Then we will introduce the notion of partial cleft coextensions which are dual to partial cleft extensions introduced by M. M. S. Alves et.al., and discuss its relation with partial crossed coproducts introduced by the first author of this paper, which covers the classical results in classical Hopf algebra theory.