Narrow Results

A new map (Λ_E) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of Λ_E is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φ_E and ψ_E) are also reviewed, in order to compare them with Λ_E. Interesting properties of the fixed points of φ_E and are studied. Among others, the fixed points of Λ_E (Fix(Λ_E)) are proved to be the maximal fuzzy points of (X, E) and the fixed points of coincide with the Image of Λ_E. An isometric embedding of X into Fix(Λ_E) is established and studied.

The aim of this paper is to introduce the idea of fuzzy betweenness relation on a set X. This is done by generalizing the definition of betweenness relation proposed by Menger. It is proved that a separating T-indistinguishability operator on X (with T a strict archimedean t-norm) generates a fuzzy betweenness relation on X and reciprocally, every fuzzy betweenness relation on X defines a separating T-indistinguishability operator on X. Moreover, it is proved that the crisp part of a fuzzy betweenness relation is a classical (metric) betweenness relation.

The main idea of this paper is the introduction of a duality between the elements of a set X and its fuzzy subsets. This duality induces a useful interpretation of the elements as fuzzy subsets and viceversa. As a natural consequence of this interpretation, the set of the fuzzy sets over [0, 1]^X can be viewed as a classical extension of X. In the case that X is endowed with a T-indistinguishability operator E and considering only the set H of the generators of E, the operator E can be naturally defined over the mentioned extension. On the other hand, a dual interpretation of the representation theorem, leads to a similar theorem but with a set of points as generators. Finally, it is also shown that T-similarities over [0,1]^H are a suitable tool in order to built a sound theoretical background for the study of the inference by similarity.