Fuzzy Closure Operators

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

“…Dilworth (see [48]). This concept is more general than the one used by Bȇlohlávek in [2]. In particular, notice that the operation • * in Remark 5 is non-commutative.…”

“…In the present paper, (L, ∧, ∨, * , * , 0, 1) will denote a residuated lattice with card L > 1 (see [2]). L will be the range of the memberships of the fuzzy subsets of X.…”

“…Both fields require the algebraic framework of complete lattices where the above mentioned operators can be seen as closure operators ( [2,3,4,47]). These operators have also been studied as fuzzy modal operators (a thorough treatment from this perspective can be found in [37]).…”

Operators and spaces associated to matrices with grades and their decompositions

“…Note that these operators and their sets of fixpoints have extensively been studied, see e.g. [1,2,4,12,18]. Note that if L = {0, 1}, B(X ↑ , Y ↓ , I) coincides with the ordinary concept lattice of the formal context consisting of X, Y , and the binary relation (represented by) I.…”

“…(1) If V ∈ L n are regarded as fuzzy sets, the concepts of an i-subspace and a c-subspace coincide with the concept of a fuzzy interior operator and a fuzzy closure operator as defined in [2,5].…”