Ordered Observation System

In paper [2] ,the point of view of observation structures and observations systems was investigated.The main result is that every nontrivial(non-identically empty) functor on the category of sets gives rise in a canonical way to a functor on the category of observation structures having a unique fixed poin.It was also shown that the resulting category of coalgebras had a final coalgebra. In paper [1] ,the notions of ordered observation structures and ordered observation systems were introduced .These were asymmetric generalizations of observation structures and observations systems discussed in paper [2].In this thesis ,we do the further study based on the notions of [1]. Firstly ,we construct separated completion of a ordered observation structure .Secondly, we extend any nontrivial set functor F to a functor F[·] on the category of ordered observation structure and to a functor F|-[·] on the category of separated and complete ordered observation system. We also prove the existence and uniqueness of fixed point of this two functor . Moreover, we naturally understand an F- coalgebra as a ordered observation system, and prove the existence of a final F[·]- coalgebra .