A note on negations and nilpotent t-norms

AbstractIn this paper, we explore general relationships among negations, convex Archimedean nilpotent t-norms, and automorphisms of the unit interval I. Each nilpotent t-norm has a (strong) negation naturally associated with it, namely, η△(x)=⋁{y∈[0,1]:x△y=0}. The same negation is determined by the formula η△(x)=f−1(f(0)/f(x)) where f is a (multiplicative) generating function for the t-norm △. A system (I,△,▿,∗) is called de Morgan if x▿y=(x∗△y∗)∗; Stone if x△y=0 if and only if y⩽x∗, and x∗▿x∗∗=1; and Boolean if it is both de Morgan and Stone. A system is shown to be Boolean if and only if ∗=η△ and x▿y=η△(η△x)△η△(y)). We also look at de Morgan, weak Boolean and Stone systems on the lattice I[2]={(x,y)∈I×I:x⩽y} and compare properties of related systems on I and on I[2]