In the literature there are two different approaches that extend the classical crisp notion of equivalence relation to the fuzzy framework. On the one hand, one can find the notion of indistinguishability operator and a few of its generalizations. These can be understood as a kind of measurement of the degree of similarity or indistinguishability between objects. On the other hand, fuzzy (quasi-)metrics measure such a degree with respect to a parameter. The study of both types of the aforesaid notions has been carried out independently without any connection between them. As a consequence, the notion of modular indistinguishability operator has been introduced recently. Such a notion unifies under the same framework both aforesaid similarity concepts. In this paper, we explore the aggregation problem for modular indistinguishability operators and for several generalizations. Hence we introduce the notions of modular fuzzy pre-order, modular fuzzy partial order and modular equality and we characterize the functions that are able to fuse all these different types of modular similarities. The aforementioned characterizations are stated in terms of triangular triplets or related notions, monotony and dominance. In contrast to the non-modular case, the class of those functions that merge modular fuzzy pre-orders (modular fuzzy partial orders) is shown to match the class of modular indistinguishability operators (modular equalities). Furthermore, the relationships between the non-modular aggregation problem, the modular one and the fuzzy metric aggregation problem are explored and the differences between them are clarified by means of appropriate examples.

In the last years, the interest in the notion of fuzzy metric has been growing in such a way that many works have focused their efforts on the study of their topological properties and their applications to Engineering problems. However, the applicability of fuzzy metrics is limited due to lack of examples in the literature. Motivated, on the one hand, by these facts and, on the other hand, by the fact that most of the instances of fuzzy metrics in the literature are constructed from classical metrics, in this paper we introduce two new techniques which allow us to construct systematically fuzzy metrics from metrics in such a way that the celebrated classical method for constructing indistinguishability operators from metrics is retrieved as a particular case. Hence, we construct strong fuzzy metrics from a given classical one considering continuous Archimedean t-norms and the pseudo-inverse of their additive generators acting on the metric modified by a positive real function. Moreover, we extend this technique tackling the particular case of the minimum t-norm, which is continuous but non-Archimedean. In such a construction, two non-negative real functions are now involved in order to modify the classical metric and one of them must be superadditive. In this case, the fuzzy metric obtained is not strong in general. Furthermore, the new methods are illustrated by means of different examples which, in addition, show that some celebrated examples of fuzzy metrics can be retrieved as a particular case through them. Finally, in the light of the developed theory, an open problem about strong fuzzy metrics is solved completing the partial solutions that can be found in the literature.

It is a well-known fact that the topology induced by a fuzzy metric is metrizable. Nevertheless, the problem of how to obtain a classical metric from a fuzzy one in such a way that both induce the same topology is not solved completely. A new method to construct a classical metric from a fuzzy metric, whenever it is defined by means of an Archimedean t-norm, has recently been introduced in the literature. Motivated by this fact, we focus our efforts on such a method in this paper. We prove that the topology induced by a given fuzzy metric M and the topology induced by the metric constructed from M by means of such a method coincide. Besides, we prove that the completeness of the fuzzy metric space is equivalent to the completeness of the associated classical metric obtained by the aforementioned method. Moreover, such results are applied to obtain fuzzy versions of two well-known classical fixed point theorems in metric spaces, one due to Matkowski and the other one proved by Meir and Keeler. Although such theorems have already been adapted to the fuzzy context in the literature, we show an inconvenience on their applicability which motivates the introduction of these two new fuzzy versions.

Response-threshold methods stand out among the different developed swarm-like methodologies that address the task allocation problem, which must be faced in multi-robot systems in order to assign to each robot the best task to perform at each instant of time. In many real missions the tasks have associated deadlines. However, the literature only contains a few swarm methodologies, and thus response-threshold methods, tackling tasks with deadlines. Motivated by this fact, in this paper, we propose a new task allocation strategy inspired by response-threshold methods which deals with tasks with time deadlines, models stimuli using fuzzy sets and, in addition, in which each robot makes the decision about the best task to perform through the celebrated Bellman-Zadeh fuzzy optimization technique. An extensive number of simulations have been carried out in order to evaluate the quantitative performance of the swarm system based on the new approach. The results confirm that the proposed mathematical approach is able to model the evolution of the system when tasks with deadlines are under consideration. We have also observed competitive performance on a fleet of real robots, which corroborates the results derived from the simulations.

Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the mipoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of “compromise” or “middle way” between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved.

We characterise those quasi-metric spaces (X, d) whose poset BX of formal balls satisfies the condition (*) From this characterisation, we then deduce that a quasi-metric space (X, d) is Smyth-complete if and only if BX is a dcpo satisfying condition (*). We also give characterisations in terms of formal balls for sequentially Yoneda complete quasi-metric spaces and for Yoneda complete T1 quasi-metric spaces. Finally, we discuss several properties of the Heckmann quasi-metric on the formal balls of any quasi-metric space.

Given a partial metric space (X,p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X,dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X,p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X,p), we construct a Smyth complete quasi-metricqonBXthat extends the quasi-metricdpsuch that both the Scott topology and the partial order ⊑dpare induced byq. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Künzi, H. Pajoohesh and M. P. Schellekens (Künziet al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.