Right Type Departmental Bulletin Paper

Let $E $ be alinear space over $\mathbb{R} $ , and $P $ be aconvex cone in $E $ satisfying (PI) $E=P-P$, (P2) $P\cap(-P)=\{0\} $. An order relation in $E $ can be defined by $x\leq y\Leftrightarrow y-x\in P $. We call alinear space $E $ equipped with such apositive cone $P $ a(partially) ordered linear space, and denote it by $(E, P) $. For asubset $A $ of $E $ , we denote the set of upper bounds and lower bounds by $\mathrm{U}\{\mathrm{A})=\{x\in E|y\leq x, \forall y\in A\} $ , $L(A)=\{x\in E|y\geq x, \forall y\in A\}$ respectively. These sets have aproperty of symmetry in the following sence. ([4]) (1) $U(L(U(A)))=U(A) $ $(A\subset E) $. In [4], the method of constructing acompletion $(\tilde{E},\tilde{P}) $ of $(E, P) $ by using the set of upper bounds $U(A) $ has been introduced. The relation (1) plays fundamental roles in the construction of $(\tilde{E},\tilde{P}) $. Also, the completion can be represented by the set of the generalized supremum in $E $ which has been introduced in [2]. We will state the summary of those results in the first part of this section. Let $\mathfrak{B} $ and $\mathfrak{B}’ $ be the fainily of all upper bounded subset and lower bounded subset in $E $ respectively, i.e. $\mathfrak{B}=\{A\subset E|A\neq\emptyset, U(A)\neq\emptyset\} $ , $\mathfrak{B}’=\{B\subset E|B\neq\emptyset, L(B)\neq\emptyset\} $. The relation