Generalized distances (partial metrics with values in quantales) and generalized equalities (quantale-valued fuzzy equalities) coincide.
The generalized distances without the axiom p(x,x)=0 are frequently used to define non-Hausdorff topologies. They have various applications in semantics of programming languages and in the theory of computations with real numbers.
Quantale-valued sets (sets with quantale-valued fuzzy equalities) were introduced by Ulrich Höhle in order to give solid foundation to the theory of fuzzy sets. They generalize Ω-sets introduced by Fourman and Scott (and also by Higgs) in connection with sheaves and logic.
In the following preprint we observe that the axioms for partial metrics with values in quantales coincide with the axioms for quantale-valued sets for commutative quantales.
Michael Bukatin, Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh. Partial Metrics and Quantale-valued Sets. Preprint, September 14, 2006.
Corrected slides for CCA 2006.
Some examples.
Published extended abstract: Michael Bukatin, Ralph Kopperman, Steve Matthews, and Homeira Pajoohesh, Partial Metrics and Quantale-valued Sets (Extended Abstract), in D.Cenzer et al, editors, CCA 2006: Proceedings of the Third International Conference on Computability and Complexity in Analysis, Informatik Berichte, 336 (09/2006), FernUniversitaet in Hagen, pp. 91-92.
Slides for "Partial Metrics, Fuzzy Equalities, and Metric-Entropy Pairs" talk on June 10, 2009.
The slides and the abstract for a talk at SumTopo 2009.
The slides for a talk at BLAST'09 conference.
The slides (1.6MB) for a On duality between metric and logical viewpoints talk at SumTopo 2011 conference. The slides are large because they include 3 pages from an unpublished 1973 manuscript by Denis Higgs. These pages construct a fuzzy equality valued in Grothendieck topology.
Published extended_abstract: Michael Bukatin, Ralph Kopperman, and Steve Matthews, On the Nature of Correspondence between Partial Metrics and Fuzzy Equalities, in the Proceedings of the 33rd Linz Seminar on Fuzzy Set Theory: Enriched Category Theory and Related Topics, February 2012.
We have published a complete research paper summarizing our understanding of this subject:
Michael Bukatin, Ralph Kopperman, and Steve Matthews, Some corollaries of the correspondence between partial metrics and multivalued equalities, Fuzzy Sets and Systems, 256, 1 December 2014, pp. 57-72.
If your library does not have access to this paper, please send me an e-mail. I hope to put an electronic version of it on this site at some point.