Projectivity and unification in substructural logics of generalized rotations

We develop a unifying approach to study projectivity and unification in substructural logics corresponding to varieties of residuated lattices generated by generalized rotation constructions. These include many interesting varieties especially in the realm of mathematical fuzzy logics. Our main results pertain what we shall call radical-determined varieties of rotations, which include all of the most relevant varieties in this framework. We characterize free algebras in a radical-determined variety of rotations in terms of weak Boolean products of rotations of free algebras in the variety of radicals, the latter being the intersections of maximal filters of the algebras in . Then we use such description to study projectivity in these varieties of rotations, characterizing finitely generated projective algebras. Moreover, we show that the strong unitary unification type of a variety of radicals implies the strong unitary type for the generated variety of rotations, which can be used to deduce the decidability of the admissibility of rules. As relevant applications of our general results, we obtain that product logic and nilpotent minimum logic have (strong) unitary unification type.