Normal forms for the one-variable fragment of Hajek´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 gives a concrete representation of the free BL-algebra BC₁ 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 to the analogous fragment of BL, giving an algorithm to express any BL-formula with one variable as a conjunction of Schauder hats.