Partial Expansion Of Partial Cross Product

In this paper, we will introduce the notion of partial cleft coextensions which are dual to partial cleft extensions introduced by Alves et.al., and discuss its relation with partial crossed coproducts introduced by the authors of this paper. This paper consists four sections.The first section is the intuoduction.In the second section, we recall some basic concepts related to twisted partial Hopf coactions and introduce the symmetric twisted partial coactions.In the third section, we discuss some properties of the symmetric twisted partial coactions.In the fourth section, we introduce the notion of partial cleft coextensions and discuss its relation with partial crossed coproducts. There is a important conclusion:Theorem 4.3 Let D be a right H-comodule coalgebra and D/C an H-coextension. Then the following are equivalent:1. The H-coextension D/C is partially cleft,2. D is isomorphic to a partial crossed coproduct C(?)(ρA,w)H with respect to a symmetric twisted partial H-comodule structure on A.