Given a loset I, every surjective map p: A --- > I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as " bubbles " (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles . As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
Given a linearly ordered set I, every surjective map p: A -- > I endows the set A with a structure of set of preferences by "replacing" the elements of I with their inverse images via p considered as " balloons " (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons . As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
R
loset
linearly ordered set
R
---
--
R
preordered set
set of preferences
R
bubbles
balloons
R
structure of loset
linear order
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bubbles. Every bubbling
balloons. Every ballooning
R
loset
linearly ordered set
R
structure of preordered set A
set of preferences whose preference relation
R
whose preorder has negatively transitive asymmetric part
is negatively transitive
R
bubbling up
ballooning
R
loset
linearly ordered set
R
bubbles
balloons
R
preordered set
set of preferences