R-partial Order On The Set Of The Results

The concept of R- poset is proposed in [8], the definition is as follows : suppose (A, (?) ) is a poset, ω is natural number set whose partial order is denoted by ≤; if R = ((?)_n)_n∈ω is a family of partial orders on A, and forall n, m∈ω, m≤n, (?)_n (?)(?)_m ,∩_n∈ω (?)_n =(?), then we call A a poset with partial order family R, R-poset for short, noted with (A, (?); R). In the paper, through interesting examples, we can show when the orders in R approximate some order, the structure of cpo( complete partial order), algebraic cpo or continuous cpo doesn't have to be preserved, and the present paper pointes out when algebraic cpo or continuous cpo can be preserved. The fixed point theorem of R-poset of R-continuous mapping is given in [7], in the present paper we show if R-continuous mapping is (Scott) continuous for every order (?)_n, then it is also ( Scott) continuous for (?), and furthermore the fixed points on the partial orders (?)_n ( n∈ω ) from partial order family exactly construct a approximating sequence for fixed point of f on the partial order which is approximated by the partial order family . Besides, we compare the scott topology on partial order from approximat -ing partial order family, and there is a simple application of this result.