Abstract
In 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] .
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