This doctoral thesis is devoted to investigate the problem of establishing
connections between Domain Theory and the theory of fuzzy metric spaces, in
the sense of Kramosil and Michalek, by means of the notion of a formal ball,
and then constructing topological and computational models for (complete)
fuzzy metric spaces.
The antecedents of this research are mainly the well-known articles of A.
Edalat and R. Heckmann [A computational model for metric spaces, Theoret-
ical Computer Science 193 (1998), 53-73], and R. Heckmann [Approximation
of metric spaces by partial metric spaces, Applied Categorical Structures 7
(1999), 71-83], where the authors obtained nice and direct links between Do-
main Theory and the theory of metric spaces - two crucial tools in the study
of denotational semantics - by using formal balls.
Since every metric induces a fuzzy metric (the so-called standard fuzzy
metric), the problem of extending Edalat and Heckmann's works to the fuzzy
framework arises in a natural way.
In our study we essentially propose two di erent approaches. For the
rst one, valid for those fuzzy metric spaces whose continuous t-norm is
the minimum, we introduce a new notion of fuzzy metric completeness (the
so-called standard completeness) that allows us to construct a (topological)
model that includes the classical theory as a special case. The second one,
valid for those fuzzy metric spaces whose continuous t-norm is greater or
equal than the Lukasiewicz t-norm, allows us to construct, among other
satisfactory results, a fuzzy quasi-metric on the continuous domain of formal
balls whose restriction to the set of maximal elements is isometric to the
given fuzzy metric. Thus we obtain a computational model for complete
fuzzy metric spaces.
We also prove some new xed point theorems in complete fuzzy metric
spaces with versions to the intuitionistic case and the ordered case, respec-
tively.
Finally, we discuss the problem of extending the obtained results to the
asymmetric framework. / Ricarte Moreno, L. (2013). Topological and Computational Models for Fuzzy Metric Spaces via Domain Theory [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34670
| Identifer | oai:union.ndltd.org:upv.es/oai:riunet.upv.es:10251/34670 |
| Date | 23 December 2013 |
| Creators | RICARTE MORENO, LUIS-ALBERTO |
| Contributors | Romaguera Bonilla, Salvador, Universitat Politècnica de València. Departamento de Matemática Aplicada - Departament de Matemàtica Aplicada |
| Publisher | Universitat Politècnica de València |
| Source Sets | Universitat Politècnica de València |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/acceptedVersion |
| Source | Riunet |
| Rights | http://rightsstatements.org/vocab/InC/1.0/, info:eu-repo/semantics/openAccess |
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