Smallest quasi-transitive extensions

Quasi-transitivity is a weakening of transitivity that has some attractive features, such as the important role it plays in the context of path-independent choice. However, the property suffers from the shortcoming that it does not allow for the existence of a closure operator. This paper examines the question to what extent an alternative operator can be defined that may then be used to ameliorate some of the limitations of quasi-transitivity imposed by the absence of a well-defined closure. To do so, we define the concept of a smallest quasi-transitive extension. A novel weakening of quasi-transitivity turns out to be necessary and sufficient for the existence of such a smallest extension. As an illustration, we apply the notion of a smallest quasi-transitive extension in the context of rational choice on arbitrary domains.