Continuous Multi-Utility Representations of Preorders and the Chipman Approach
Abstract
:1. Introduction
1.1. The Chipman Approach
- SB:
- In order for to be Hausdorff, it is necessary that the sets and , where x runs through X, constitute a sub-basis of .
- LR:
- In order for to be Hausdorff, it is necessary that, for all points and , the validity of the following implication holds for all and for all :
1.2. The Continuous Multi-Utility Representation Theorem
2. On an Order-Embedding Theorem of Chipman-Type
- (i)
- There exists some cardinal number κ for which there exists an order-embedding .
- (ii)
- There exists some cardinal number κ and a (complete) preorder ≲ on , that is coarser than , for which there exists a continuous order-embedding .
- (i)
- There exist some ordinal number λ and order-embeddingsandsuch that the following diagram commutes
- (ii)
- In addition to Assertion(i), (total) preorders on D and and order-embeddingsandcan be chosen in such a way that the compositionsandare continuous and the following diagram commutes
3. On a Relation of the Chipman Approach with the Continuous Multi-Utility Representation Problem of Preorders
- HD:
- Let ≾ have a continuous multi-utility representation. Then, is a Hausdorff space.
- OC:
- Let be a Hausdorff space and let ≾ be closed. Then, is open (and closed) for every point that is maximal with respect to .
- (i)
- ≾ admits a continuous multi-utility representation.
- (ii)
- is Hausdorff and ≾ is closed.
- (iii)
- is Hausdorff and ≾ is semi-closed.
- (i)
- There exists some cardinal number κ and a preorder ≲ on , which is coarser than , for which there exists a continuous order-embedding ;
- (ii)
- ≾ admits a continuous multi-utility representation.
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Bosi, G.; Daris, R.; Zuanon, M. Continuous Multi-Utility Representations of Preorders and the Chipman Approach. Axioms 2024, 13, 148. https://doi.org/10.3390/axioms13030148
Bosi G, Daris R, Zuanon M. Continuous Multi-Utility Representations of Preorders and the Chipman Approach. Axioms. 2024; 13(3):148. https://doi.org/10.3390/axioms13030148
Chicago/Turabian StyleBosi, Gianni, Roberto Daris, and Magalì Zuanon. 2024. "Continuous Multi-Utility Representations of Preorders and the Chipman Approach" Axioms 13, no. 3: 148. https://doi.org/10.3390/axioms13030148
APA StyleBosi, G., Daris, R., & Zuanon, M. (2024). Continuous Multi-Utility Representations of Preorders and the Chipman Approach. Axioms, 13(3), 148. https://doi.org/10.3390/axioms13030148