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 das (strong) negation naturally associated with it, namely, eta(triangle)(x) = V{y is an element of [0, 1] : x Delta y = 0}. The same negation is determined by the formula eta(triangle)(x) = f(-1) (f (0)/f(x)) where f is a (multiplicative) generating function for the t-norm triangle. A system (I, triangle, del, *) is called de Morgan if x del y (x* triangle y*)*; Stone if x triangle y = 0 if and only if y less than or equal to x*, and x*del x** = 1; and Boolean if it is both de Morgan and Stone. A system is shown to be Boolean if and only if * = eta(triangle) and x del y = eta(triangle)(eta(triangle)(x) (triangle) eta(triangle)(y)) We also look at de Morgan, weak Boolean and Stone systems on the lattice I-[2] = {(x, y) is an element of I x I : x less than or equal to y} and compare properties of related systems on I and on I-[2]. (C) 1999 Elsevier Science Inc. All rights reserved.