Standard theories of fuzzy sets with the law (μ∧σ′)′=σ∨(μ′∧σ′) Original Research Article

It is known that, in standard theories of fuzzy sets ([0,1]X,∧,∨,′), the law (μ∧σ′)′=σ∨(μ′∧σ′) does not hold if ∧ and ∨ are dual. It is also known that considering, in both sides of this formula, different t-norms and negation functions, there are uncountable many solutions of the equivalent functional equation N1(T1(a,N2(b)))=S(b,T2(N3(a),N4(b))) in the unknowns N1, N2, N3, N4, T1, T2, and S. Nevertheless, since the simplest situation in which N1=N2=N3=N4, T1=T2, remained open, this paper is devoted to completely solve this particular case. That is, to study in which standard theories of fuzzy sets ([0,1][0,1],T,S,N) the above law holds. The solution is that the law only holds in the theories isomorphic to ([0,1]X,Prod,W∗,1−id[0,1]). This opens the door to consider nondual standard theories of fuzzy sets, a field until today largely ignored.