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<p>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 <i>strong external actions</i>. 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.&#xa0;Rump in the cateogory of L-algebras is established.</p>

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On Actions and Split Extensions in Varieties of Hoops: The Case of Strong Section

  • 期刊论文

  • 发表日期：

    2026年6月24日

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  • Manuel Mancini,
  • Giuseppe Metere,
  • Federica Piazza

摘要

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.