Facial structure of convex sets and integrand representation of convex operators

对于有序线性空间E中的子集A，广义上确界Supa被定义为U(A)的所有极小元素的集合(所有上界的总和)。关于广义上确界，已经得到了许多有趣的结果，这可以应用于集合优化理论等。设X是2^E关于等价关系A~B⇔U(A)=U(B)(A，B⊂E)的商集.在E不是序完备(或格)的情况下，通过定义向量运算和X的自然序，我们发现X成为序完备向量格，并且X有一个序同构于E的子空间.此外，我们还可以看到，在自然条件U(A)=(Supa)+P(P：E中的正锥)下，X可以用E中所有广义上确界的集合来标识.2000年8月在京都举行的“非线性分析和凸性分析研究”会议上报告了这些结果。序线性空间(E，P)称为单调序完备(m.o.c.)如果每个全序子集A⊂E具有U(A)≠φ有最小上界。在处理广义上确界时，单调序完备性和P(P的面结构)的一些几何性质以及U(A)=(Supa)+P的一些几何性质起着重要的作用。例如，如果正锥P是代数闭的，且P的每个面都是有限维的，则条件U(A)=(SUPA)+P成立。通过构造一个例子，我们还证明了反之亦然。此外，我们还证明了P的代数闭性是U(A)=(Supa)+P条件的必要条件。我们准备发表这些结果。此外，这项研究的主要结果将在2001年7月举行的国际会议“NACA 2001”上报告。

For a subset A in an ordered linear space E, the generalized supremum SupA is defined as the set of all minimal elements of U (A)(the totality of all upper bounds). Many interesting results about the generalized supremum has been obtained so for, and this can be applied to the theory of set optimization for example. Let X be the quotient set of 2^E with respect to the equivalence relation A〜B⇔U (A)=U (B) (A, B⊂E).In the case when E is not order complete (or a lattice), we have found that X becomes an order complete vector lattice by defining a vector operation and a natural order to X and that X has a subspace which is order isomorphic to E.Moreover we can see that X can be identified with the set of all generalized supremum in E, under the natural condition U (A)=(SupA)+P (P : positive cone in E). These results was reported at the conference "Research in Nonlinear Analysis and Convex Analysis" which was held at Kyoto in August 2000. An ordered linear space (E, P) is said to be monotone order complete (m.o.c.) if every totally ordered subset A⊂E with U (A)≠φ has the least upper bound. When we deal with the generalized supremum, the monotone order completeness and some geometric properties of P (facial structure of P) play important roles as well as the condition U (A)=(SupA)+P.In this research we have obtained some relations between these conditions. For example, if the positive cone P is algebraically closed and every face of P is finite dimensional, then the condition U (A)=(SupA)+P holds. By constructiong an example, we have also proved that the converse does not true. Moreover, we have proved that the algebraic closedness of P is necessary to the condition U (A)=(SupA)+P.We are preparing to publish these results. Also, the main results in this research will be reported at the international conference "NACA 2001" which is held in July 2001.