Some representable De Morgan algebras

Abstract

The Stone theorems give us a representation of Boolean algebras, the structure associated to classical logic, through set algebras. The present paper deals with the representation of De Morgan algebras, one of the most important structures associated to multiple-valued logic, through fuzzy set algebras. Let us recall that a De Morgan algebra (9, A, V, n) is a structure such that (Pi, A, V) is a distributive lattice with supremum u and infimum 0, and n is a decreasing involution in 9 such that n(0) = u and n(u) = 0 and, consequently, satisfies the De Morgan laws. We call these mappings n: 9 -+ 9 strong negations or involutions of 40 (see, for instance, [8]). On the other hand, let us recall that Zadeh [3] defined the fuzzy subset of a universe X being characterized by a mapping A : X+ [0, 11, the generalized characteristic function, which sends each x E X into the “membership grade” of x belonging to A. As usual, we denote by p(X) the set of fuzzy subsets of x, f(X) = [O, 11*, and by Q’(X), n, U) the lattice obtained by defining f7 and U between fuzzy sets through (A ~7 B)(x) = min(A(x), B(x)) and (A U B)(x) = max(A (x), B(x)). This lattice is distributive with supremum X (defined by X(x) = 1 for all x E X), infimum 0 (defined by 0(x) = 0 for all x E X) and such that the Boolean algebra of the boolean elements of f(X) is the Boolean algebra of the classical subsets of X, (P(X), n, U, C). The concept of fuzzy subset of a universe X is generalized (Goguen [4]) by defining fuzzy subsets taking values in a lattice (L, A, V). It can still be more widely generalized by defining fuzzy subsets that for all x E X take values in different lattices. Thus, if 9’ = {R, 1 x E X} is a family of lattices (R,, A,,V,) with supremum 1, and inlimum O,, we can define a fuzzy subset of X taking values in 5%’ as any mapping A: X-1 U,,, R, such hat A(x) E R, for all x E X. If we denote by P&X) the set of fuzzy subsets of X taking values in .R and define n and U through (A n B)(x) = A (x) A, B(x) and (A U B)(x) = A(x) V, B(x), then (P&X), n, U) happens to be a lattice 463 0022.247)(/84$3.00