Directed Completions and DM-Completions on $\mathcal {R}-$Posets

Abstract

As an enrichment of single order the notion of $\mathcal{R}$-posets have been introduced by Ji Wen in [8]. Let $(P,\sqsubseteq)$ be a poset and ω the natural number set whose order is denoted by ≤. If $\mathcal {R}=(\sqsubseteq_n)_{n\in\omega}$ is a family of partial orders on P such that (i)∀ n,m ∈ ω,m ≤ n implies $\sqsubseteq_n \subseteq \sqsubseteq_m$, and (ii)${\bigcap}_{n\in\omega}\sqsubseteq_n=\sqsubseteq$, then call $(P,\sqsubseteq)$ a poset with partial order family $\mathcal {R}$($\mathcal {R}$-poset for short),denoted by $(P,\sqsubseteq;\mathcal {R})$. It provides possibility to interpret or measure the complex information in stepwise computing. We will write simply $P_n=(P,\sqsubseteq_n)$ and $P=(P,\sqsubseteq)$ respectively if no confusion can rise.

Let $\mathcal {C}(P_n)$ be some completion of P _n .It is of interest to know the connections between $\mathcal {C}(P_n)$ and $\mathcal {C}(P_{n+1})$ or between $\mathcal {C}(P_n)$ and $\mathcal {C}(P)$. Let $\mathcal {O}(P)(\mathcal {O}(P_n))$ and $\mathcal {I}(P)(\mathcal {I}(P_n))$ denote all lower sets and all ideals of P(P _n ) respectively.It concludes that $\mathcal {O}(P_n)\subseteq\mathcal {O}(P_{n+1})\subseteq \mathcal {O}(P)$ for all n ∈ ω which implies that $\bigcup_{n \in\omega}\mathcal {O}(P_n)\subseteq\mathcal {O}(P)$.But if we require that $\mathcal {I}(P_n)\subseteq\mathcal {I}(P_{n+1})\subseteq \mathcal {I}(P)$ and =  _n  is  =  _P   for all n ∈ ω then $\sqsubseteq_n=\sqsubseteq$ for all n ∈ ω. The Dedekind-MacNeile completions(DM-completion for brevity) are also investigated.It is concluded that $\sqsubseteq_n=\sqsubseteq$ for all n ∈ ω if $\sqsubseteq_{n+1}$ is a close subrelation of $\sqsubseteq_n$ for all n ∈ ω. The Glois connections can be well preserved on every order in $\mathcal {R}-$poset.

It is worth pointing out that several interesting examples are indicated to make the study more intelligible.It is our future work to apply domain theory to the formal concept analysis,where it will be possible to approximate infinite informations by finite or computable ones.