Abstract
The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to merge an arbitrary family of (bases of) L-probabilistic quasi-uniformities into a single one. These fuzzy structures are special filters of fuzzy binary relations. Hence we first make a complete study of functions between partially-ordered sets that preserve some special sets, such as filters. Afterwards, a complete characterization of those functions aggregating bases of L-probabilistic quasi-uniformities is obtained. In particular, attention is paid to the case L={0,1}, which allows one to obtain results for functions which aggregate crisp quasi-uniformities. Moreover, we provide some examples of our results including one showing that Lowen’s functor ι which transforms a probabilistic quasi-uniformity into a crisp quasi-uniformity can be constructed using this aggregation procedure.
Highlights
Aggregation can be considered as a method for merging a list of numbers in a single representative one
This paper is devoted to examining the aggregation of another important fuzzy structure: The L-probabilistic quasi-uniformities
If we intend to characterize those functions merging a family of probabilistic quasi-uniformities into a single one, it is natural to guess that the aggregation of fuzzy binary relations will have an important role
Summary
Aggregation can be considered as a method for merging a list of numbers in a single representative one. Aggregation functions based on bounded partially ordered sets rather than on ([0, 1], ≤) have been studied [5,6] In mathematics, this process of aggregation does not always consider numbers but other mathematical structures. This paper is devoted to examining the aggregation of another important fuzzy structure: The L-probabilistic quasi-uniformities In this way, the main goal of this paper is to characterize those functions, which allow the merging of an arbitrary family of L-probabilistic quasi-uniformities into a single one (see Definitions 6, 7 and 9). If we intend to characterize those functions merging a family of probabilistic quasi-uniformities into a single one, it is natural to guess that the aggregation of fuzzy binary relations will have an important role. We show that the construction of a crisp quasi-uniformity starting from a probabilistic quasi-uniformity by means of the Lowen’s functor ι [29] can be performed with our results
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