Fuzzy derivations (Q1369323)

The authors introduce the concept of a fuzzy derivation as a natural extension of the basic concept of a derivation in Galois theory. Here it is defined as a special type of fuzzy function satisfying the chain rule, i.e. the sup-min composition of two fuzzy functions is again a fuzzy function. A fuzzy set \(T\) in the Cartesian product of two sets \(A\) and \(B\) is called a fuzzy function iff for every \(x\) in \(A\) the cardinality of the support of the \(T\)-afterset of \(x\) is smaller than or equal to 1. It is shown that a fuzzy derivation is a fuzzy linear operator and that the Lie commutator of two fuzzy derivations is again a fuzzy derivation. Furtheron the authors introduce the algebra of dual numbers over an algebra over a given field and they prove that a fuzzy derivation can be determined by a fuzzy isomorphism of this algebra of dual numbers. An element \(c\) of \(A\) is called a \(D\)-constant where \(D\) is a fuzzy derivation of a subalgebra \(A\) of an algebra \(B\) iff \(D(c,0)>0\). It is shown that the set of \(D\)-constants constitutes a subalgebra of \(A\). In a quite natural way it is shown that every fuzzy derivation induces a crisp derivation. The paper ends with some additional properties of fuzzy derivations.