Towards probabilistic partial metric spaces: Diagonals between distance distributions

Abstract

The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the quantaloid of diagonals between distance distributions, which is expected to establish the categorical foundation of probabilistic partial metric spaces. Observing that the quantale of distance distributions w.r.t. an arbitrary continuous t-norm is non-divisible, we precisely characterize diagonals between distance distributions, and prove that one-step functions are the only distance distributions on which the set of diagonals coincides with the generated down set.for all t ∈ [0, ∞], called a distance distribution, rather than a non-negative real number. As discovered by Chai [4] and investigated by Hofmann-Reis [12], (generalized) probabilistic metric spaces are categories enriched in the quantale ∆ * = (∆, ⊗ * , κ 0,1 ) of distance distributions w.r.t. a continuous t-norm * on the unit interval [0, 1] (see Proposition 3.6 for further explanations) which, in elementary words, are precisely sets X equipped with a map α :for all x, y, z ∈ X and r, s ∈ [0, ∞]. Partial metric spaces are metric spaces in which the distance from a point to itself may not be zero. Explicitly, (generalized) partial metric spaces are sets X equipped a with a map α :for all x, y, z ∈ X. As discovered by Höhle-Kubiak [17] and Pu-Zhang [29] and formalized later by Stubbe [35] in a more general setting, although partial metric spaces are not categories enriched in the quantale [0, ∞] + , one may construct a quantaloid of diagonals of the quantale [0, ∞] + , usually denoted by D[0, ∞] + , such that partial metric spaces are precisely categories enriched in the quantaloid D[0, ∞] + .It is now natural to ask whether it is possible to consider the probabilistic version of partial metric spaces or, equivalently, the partial version of probabilistic metric spaces. As far as we know, this topic has been considered recently by several authors under the name probabilistic partial metric spaces or fuzzy partial metric spaces; see, e.g. [1,31,37,38,39]. Following the categorical interpretation of partial metric spaces, this paper aims at a full-scale investigation of the quantaloid D∆ * of diagonals of the quantale ∆ * of distance distributions w.r.t. a continuous t-norm * on [0, 1], so that a categorical foundation of probabilistic partial metric spaces can be established, which was more or less neglected in the existing references.In general, let Q = (Q, &, 1) be a commutative and integral quantale (i.e., complete residuated lattice), in which p & q ≤ r ⇐⇒ p ≤ q → r for all p, q, r ∈ Q. With DQ denoting the quantaloid of diagonals of Q (see Proposition 2.2), a DQ-category (also called partial Q-category, see [14]) consists of a set X and a map α : X × X

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