In this paper we provide a concept of fuzzy partial metric space (X, P, *) as an extension to fuzzy setting in the sense of Kramosil and Michalek, of the concept of partial metric due to Matthews. This extension has been defined using the residuum operator → * associated to a continuous t-norm * and without any extra condition on *. Similarly, it is defined the stronger concept of GV-fuzzy partial metric (fuzzy partial metric in the sense of George and Veeramani). After defining a concept of open ball in (X, P, *), a topology TP on X deduced from P is constructed, and it is showed that (X, TP) is a T0-space.

In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that ⋀ t > 0 M ( x , y , t ) > 0 . The continuous extension M 0 of M to X 2 × 0 , + ∞ is called extended fuzzy metric. We prove that M 0 generates a metrizable topology on X, which can be described in a similar way to a classical metric. M 0 can be used for simplifying or improving questions concerning M; in particular, we expose the interest of this kind of fuzzy metrics to obtain generalizations of fixed point theorems given in fuzzy metric spaces.

In this paper we present an example of a fuzzy ψ-contractive sequence in the sense of D. Mihet, which is not Cauchy in a fuzzy metric space in the sense of George and Veeramani. To overcome this drawback we introduce and study a concept of strictly fuzzy contractive sequence. Then, we also make an appropriate correction to Lemma 3.2 of [V. Gregori and J. Miñana, On fuzzy ψ-contractive sequences and fixed point theorems, Fuzzy Sets and Systems 300 (2016), 93-101].

In 1994, Matthews introduced the notion of partial metric and established a duality relationship between partial metrics and quasi-metrics defined on a set X. In this paper, we adapt such a relationship to the fuzzy context, in the sense of George and Veeramani, by establishing a duality relationship between fuzzy quasi-metrics and fuzzy partial metrics on a set X, defined using the residuum operator of a continuous t-norm ∗. Concretely, we provide a method to construct a fuzzy quasi-metric from a fuzzy partial one. Subsequently, we introduce the notion of fuzzy weighted quasi-metric and obtain a way to construct a fuzzy partial metric from a fuzzy weighted quasi-metric. Such constructions are restricted to the case in which the continuous t-norm ∗ is Archimedean and we show that such a restriction cannot be deleted. Moreover, in both cases, the topology is preserved, i.e., the topology of the fuzzy quasi-metric obtained coincides with the topology of the fuzzy partial metric from which it is constructed and vice versa. Besides, different examples to illustrate the exposed theory are provided, which, in addition, show the consistence of our constructions comparing it with the classical duality relationship.