A note on negations and nilpotent t-norms

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, 4 (x) = W fy 2 [0; 1] : x4 y = 0g. The same negation is determined by the formula 4 (x) = f 1 (f (0) =f (x)) where f is a (multiplicative) generating function for the t-norm 4. A system (I;4;5; ) is called de Morgan if x 5 y = (x 4 y ) ; Stone if x 4 y = 0 if and only if y x , and x 5 x = 1; and Boolean if it is both de Morgan and Stone. A system is shown to be Boolean if and only if = 4 and x5 y = 4( 4(x)4 4(y)). We also look at de Morgan, weak Boolean and Stone systems on the lattice I[2] = f(x; y) 2 I I : x yg and compare properties of related systems on I and on I[2].