Normal Forms for the One-Variable Fragment of Hájek's Basic Logic

The variety of BL-algebras constitutes the algebraic semantic counterpart of Hajek's Basic Logic BL, that is, the infinite-valued logic of all continuous t-norms and their residua. Montagna [6] gives a concrete representation of the free BL-algebra BL1 over one generator as an algebra of piecewise linear functions. In this paper we extend Mundici's approach to normal forms for the one-variable fragment of Lukasiewicz logic [7] to the analogous fragment of BL, giving an algorithm to express any BL-formula with one variable as a conjunction of Schauder hats.