Relaxation. Coalgebra Invariant

In[1],Bart Jacobs and Jasse Hughes have brought in a new kind of functor.They took the order on a functor as a new functor.Based on that,they defined and researched some new notions about bisimulation.We take this new functor into the research of invariant in coalgebra,get the definition of predicate invariant,then we define and research several new notions.In the last,we can reach some conclusion about invariant.It is worth pointing out that we find the sufficient condition to make two-way lax invariant and invariant coincide,and prove that the great lax invariant is exactly the largest fixed point of some special functor coalgebra in set category.