New Results on the Aggregation of Norms

Abstract

It is a natural question if a Cartesian product of objects produces an object of the same type. For example, it is well known that a countable Cartesian product of metrizable topological spaces is metrizable. Related to this question, Borsík and Doboš characterized those functions that allow obtaining a metric in the Cartesian product of metric spaces by means of the aggregation of the metrics of each factor space. This question was also studied for norms by Herburt and Moszyńska. This aggregation procedure can be modified in order to construct a metric or a norm on a certain set by means of a family of metrics or norms, respectively. In this paper, we characterize the functions that allow merging an arbitrary collection of (asymmetric) norms defined over a vector space into a single norm (aggregation on sets). We see that these functions are different from those that allow the construction of a norm in a Cartesian product (aggregation on products). Moreover, we study a related topological problem that was considered in the context of metric spaces by Borsík and Doboš. Concretely, we analyze under which conditions the aggregated norm is compatible with the product topology or the supremum topology in each case.

Aggregation of Weak Fuzzy Norms

“…A similar study for asymmetric norms was developed by Martín, Mayor and Valero [8]. Recently, Pedraza and Rodríguez-López [9] have addressed the problem of the aggregation of norms on the same set.…”

“…This question for metrics has been also considered recently by Mayor and Valero [4]. Therefore, we have two different but related problems, which gave rise to two different families of functions that we next define using the terminology of [9,17,20].…”

“…Observe that, in the crisp context, the norm aggregation functions on products also coincide with the asymmetric norm aggregation functions on products [8,9]. Nevertheless, these functions are different from the (quasi-)metric aggregation functions on products [2,3].…”

“…On the other hand, if you consider (asymmetric) normed vector spaces rather than (quasi-)metric spaces, the concepts of (asymmetric) norm aggregation function on products and (asymmetric) norm aggregation function on sets can be considered in a natural manner. The former has been characterized in [7,8], while the latter has been studied in [9]. As the main objective of this paper is to study this problem in the fuzzy context, we recall the known results for crisp (asymmetric) norms.…”

“…Actually, it will be shown that weak fuzzy (quasi-)norm aggregation functions are exactly the fuzzy (quasi-)metric aggregation functions. Notice that this is not true in the crisp context where norm aggregation functions are metric aggregation functions, but the reciprocal implication does not hold in general [2,3,8,9].…”