Facial structure of convex sets and integrand representation of convex operators

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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.