Functorial relationships between Lattice-valued topology and topological systems

This paper investigates functorial relationships between lattice-valued topology (arising from fuzzy  sets and fuzzy logic) and topological systems (arising from topological and localic aspects of domains and finite observational logic in computer science). Two such relationships are embeddings from TopSys into Loc-Top, both having two fold significance: for computer science the significance is that TopSys is not topological over Set × Loc, yet Loc-Top is topological over Set × Loc hence these embeddings can be used to construct in Loc-Top the unique initial [final] lifts of all forgetful functor structured sources [sinks] in TopSys; and for topology, the significance is that both embeddings generate anti-stratified topological spaces from ordinary topological spaces and spatial locales rewritten as topological systems, thus justifying the current structural axioms of Loc-Top and lattice-valued topology (which include all anti-stratified, non-stratified, and stratified spaces). Quaestiones Mathematicae 32(2009), 139–18

Gradual elements in a fuzzy set

International audienceThe notion of a fuzzy set stems from considering sets where, in the words of Zadeh, the “transition from nonmembership to membership is gradual rather than abrupt”. This paper introduces a new concept in fuzzy set theory, that of a gradual element. It embodies the idea of fuzziness only, thus contributing to the distinction between fuzziness and imprecision. A gradual element is to an element of a set what a fuzzy set is to a set. A gradual element is as precise as an element, but the former is flexible while the latter is fixed. The gradual nature of an element may express the idea that the choice of this element depends on a parameter expressing some relevance or describing some concept. Applications of this notion to fuzzy cardinality, fuzzy interval analysis, fuzzy optimization, and defuzzification principles are outlined