On actions and split extensions in varieties of hoops

BL-algebras provide the algebraic semantics of Basic Logic, the logic of continuous t-norms, capturing the common fragment of Lukasiewicz, Gödel, and Product logics. It is well known that, up to isomorphism, every continuous t-norm behaves locally as one of the three fundamental ones: the Lukasiewicz t-norm, the Gödel t-norm, or the product t-norm. Each t-norm induces a residuation defined. The residuations associated with the three fundamental continuous t-norms were studied by P. Hájek, who provided axiomatizations of the corresponding varieties of algebras: the variety MVAlg of MV-algebras forms the algebraic semantics of Lukasiewicz Logic; the variety GAlg of Gödel algebras forms the algebraic semantics of Gödel Logic; and the variety PAlg of product algebras forms the algebraic semantics of Product Logic. Finally, P. Ha2jek introduced the variety BLAlg of BL-algebras, which provides the algebraic semantics of Basic Logic. From a categorical point of view, the variety BLAlg is an ideally exact category. The aim of this talk is to study internal actions and split extensions in the variety of hoops, with particular attention to split extensions with strong section. Such extensions are described in terms of strong external actions, i.e., a pair of maps satisfying a set of identities related to the axioms satisfied by the hoop. We prove that for any hoop X there is a natural isomorphism between the functor of strong external actions on X and the functor of isomorphism classes of split extensions with strong section with kernel X. We also show that this notion trivializes for MV-algebras, while in the variety of Gödel hoops strong external actions coincide with those of basic hoops. This is joint work with Giuseppe Metere, Federica Piazza and Marco Elio Tabacchi.