Lattices of fuzzy sets and bipolar fuzzy sets, and mathematical morphology

A Historical Account of Types of Fuzzy Sets and Their Relationships

“…In general, they work very well in problems in which we have to represent the difference between the positive and the negative representation of something [37], in particular in cognitive psychology and medicine [19]. Also in image processing they have been used often, as in [20], [89]. We should remark that the mathematical identity between these sets and IVFSs makes that, in many applications in which IVFSs are useful, so are AIFSs [41].…”

On the relation between fuzzy closing morphological operators, fuzzy consequence operators induced by fuzzy preorders and fuzzy closure and co-closure systems

“…As it can be seen in the literature, morphological fuzzy dilation and erosion use a unary structuring element together with a translation operation (see for instance equations (7) in definition 2, page 2004 in [8]). Both dilation and erosion operators are related to the set of translations in X and therefore, they are also related to the binary operation + in R For any structuring element B ⊂ X, we define the binary relation R B ⊂ X × X as follows: xR B y ⇐⇒ y ∈ B x .…”

“…Generalizations of fuzzy mathematical morphology from different perspectives and interesting applications have been recently proposed (see for instance [7,8,16,43,46]). …”

Logic for physical space

“…This framework makes the theory applicable to many different contexts as soon as a lattice structure can be defined: sets and functions (the most classical use of MM relies on the lattice of powerset with set-theoretic inclusion, and on the lattice of functions with the usual partial order), logic, and (bipolar) fuzzy sets, Bloch and Maître (1995);Bloch (2006Bloch ( , 2007Bloch ( , 2011. Note that all these formal settings are interesting for dealing with physical space from either a quantitative point of view (sets and functions), a semi-quantitative point of view, taking into account spatial imprecision (fuzzy sets), or a qualitative and symbolic point of view (logic).…”