On split extensions of product hoops

The aim of this article is to investigate internal actions/split extensions in the category of product hoops, with a focus on those that have a strong section. We provide a characterization of split extensions that strongly split in terms of strong external actions, highlighting their relevance in the categorical study of algebraic structures. Product hoops, together with their bounded counterpart, product algebras, play a significant role in the algebraic study of fuzzy logic, particularly in modeling implications and graded truth values, and the double negation retraction provides a significant example of split extensions with a strong section, thus motivating this work.