The Hausdorff fuzzy quasi-metric

Removing the condition of symmetry in the notion of a fuzzy (pseudo)metric, in Kramosil and Michalek's sense, one has the notion of a fuzzy quasi-(pseudo-)metric. Then for each fuzzy quasi-pseudo-metric on a set X we construct a fuzzy quasi-pseudo-metric on the collection of all nonempty subsets of X, called the Hausdorff fuzzy quasi-pseudo-metric. We investigate several properties of this structure and present several illustrative examples as well as an application to the domain of words. The notion of Hausdorff fuzzy quasi-pseudo-metric when quasi-pseudo-metric fuzziness is considered in the sense of George and Veeramani is also discussed.