A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set

Abstract

By analyzing proofs of the classical Riesz-Kantorovich theorem, the Mazón-Segura de León theorem on abstract Uryson operators and the Pliev-Ramdane theorem on C-bounded orthogonally additive operators on Riesz spaces, we find the most general (to our point of view) algebraic structure, which we call a complementary space, for which the theorem can be generalized with a similar proof. By a complementary space we mean a PO-set $G$ with a least element $0$ such that every order interval $[0,e]$ of $G$ with $e \neq 0$ is a Boolean algebra with respect to the induced order. There are natural examples of complementary spaces: Boolean rings, Riesz spaces with the lateral order. Moreover, the disjoint union of complementary spaces is a complementary space. Our main result asserts that, the set of all additive (in certain sense) functions from a complementary space to a Dedekind complete Riesz space admits a natural Dedekind complete Riesz space structure, described by formulas which are close to the classical Riesz-Kantorovich ones. This theorem generalizes the above mentioned Mazón and Segura de León and Pliev-Ramdane theorems. In the final section, we construct a model of market with an arbitrary commodity set, connected to a complementary space.