Open Access
Partial representations of orderings
(World Scientific, 2018) Bosi, Gianni; Estevan Muguerza, Asier; Zuanon, Magali; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability, … ) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.
Open Access
Topologies for semicontinuous Richter–Peleg multi-utilities
(Springer, 2020) Bosi, Gianni; Estevan Muguerza, Asier; Raventós Pujol, Armajac; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Institute for Advanced Research in Business and Economics - INARBE; Estadística, Informática y Matemáticas
The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition fails to be sufficient. Instead of search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements.
Open Access
Continuous representations of interval orders by means of two continuous functions
(Springer, 2020) Bosi, Gianni; Estevan Muguerza, Asier; Institute for Advanced Materials and Mathematics - INAMAT2
In this paper, we provide a characterization of the existence of a representation of an interval order on a topological space in the general case by means of a pair of continuous functions, when neither the functions nor the topological space are required to satisfy any particular assumptions. Such a characterization is based on a suitable continuity assumption of the binary relation, called weak continuity. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.

Estevan Muguerza, Asier

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