Hoops and Fuzzy Logic

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek´s Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsegood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom $(\forall 3)$ in these predicate logics is studied. Finally, computational complexity issues of the propositional logics are also addressed.