A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set
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...
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Ivan Franko National University of Lviv
2024-12-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/565 |
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| author | M. M. Popov O. Z. Ukrainets |
| author_facet | M. M. Popov O. Z. Ukrainets |
| author_sort | M. M. Popov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d56ab78b3cc7473690911c1fa9e40fc4 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-12-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-d56ab78b3cc7473690911c1fa9e40fc42025-08-20T03:33:17ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-12-0162219921010.30970/ms.62.2.199-210565A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity setM. M. Popov0O. Z. Ukrainets1Institute of Exact and Technical Sciences, Pomeranian University S{\l}upsk, Poland; Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, UkraineYurii Fedkovych Chernivtsi National University, Chernivtsi, UkraineBy 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/565riesz-kantorovich theoremriesz spaceorthogonally additive operatorarrow-debreu model |
| spellingShingle | M. M. Popov O. Z. Ukrainets A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set Математичні Студії riesz-kantorovich theorem riesz space orthogonally additive operator arrow-debreu model |
| title | A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set |
| title_full | A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set |
| title_fullStr | A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set |
| title_full_unstemmed | A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set |
| title_short | A maximal Riesz-Kantorovich theorem with applications to markets with an arbitrary commodity set |
| title_sort | maximal riesz kantorovich theorem with applications to markets with an arbitrary commodity set |
| topic | riesz-kantorovich theorem riesz space orthogonally additive operator arrow-debreu model |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/565 |
| work_keys_str_mv | AT mmpopov amaximalrieszkantorovichtheoremwithapplicationstomarketswithanarbitrarycommodityset AT ozukrainets amaximalrieszkantorovichtheoremwithapplicationstomarketswithanarbitrarycommodityset AT mmpopov maximalrieszkantorovichtheoremwithapplicationstomarketswithanarbitrarycommodityset AT ozukrainets maximalrieszkantorovichtheoremwithapplicationstomarketswithanarbitrarycommodityset |