On actions and split extensions in varieties of hoops: the case of strong section
Manuel Mancini, Giuseppe Metere, Federica Piazza
TL;DR
This work develops a categorical understanding of internal actions and split extensions in hoops and their subvarieties by introducing strong section split extensions and establishing a natural bijection with strong external actions: $\SplExt_{ss}(-,X) \cong \EAct_{ss}(-,X)$. It provides explicit semidirect product constructions in hoops, specializes the framework to basic, Wajsberg, Gödel, and product hoops, and demonstrates how these correspondences respect the respective subvariety structures. The study also connects to W. Rump's semidirect product in L-algebras, showing how hoop actions induce related semidirect products within that setting. The results yield a unified algebraic and categorical description of actions in hoop-based logics, with trivialization in MV-algebras and open questions for WHoops, guiding future extensions to broader split extensions and external-action theories.
Abstract
The aim of this article is to investigate internal actions and split extensions in the variety of hoops. We provide a characterization of split extensions with strong section in terms of strong external actions. Beyond the general setting of hoops, the study is extended to the subvarieties of basic hoops, Wajsberg hoops, Gödel hoops and product hoops. Within the setting of basic hoops and their bounded counterparts, BL-algebras, the double negation yields a significant example of split extension with strong section, thus motivating our approach. A connection between strong external actions of hoops and the semidirect product construction introduced by W. Rump in the cateogory of L-algebras is established.