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Modal weak Kleene logics: algebraic semantics and Jónsson-Tarski representation

  • Stefano Bonzio
  • Gandolfo Vergottini
  • Nicolò Zamperlin
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Abstract

In this paper we continue the work started in [Bonzio & Zamperlin 2025] by investigating the algebraic semantics for modal external weak Kleene logics. In order to do so we study the global modal logics associated with the three-valued Kripke models introduced in the previous article, showing how these systems are algebraizable. We proceed by defining their algebraic counterparts, obtaining the classes of modal Bochvar and modal Halldén algebras, whose structure theory we investigate via Płonka sum decomposition. Of particular interested is the interaction of the modal operator with the homorphisms of the Płonka decomposition. These results allow us to prove representation theorems for these expanded modal algebras in the style of the Jónsson-Tarski representation for Boolean algebras with operators. As a consequence, we also obtain algebraic semnatics for the local modal logics from which the work originated.

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Cite

BibTeXRIS
BibTeX
@misc{bonzio2026modal,
  title = {{Modal weak Kleene logics: algebraic semantics and Jónsson-Tarski representation}},
  author = {Stefano Bonzio and Gandolfo Vergottini and Nicolò Zamperlin},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.09566},
  eprint = {2611.09566},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.09566},
}
RIS
TY  - UNPB
TI  - Modal weak Kleene logics: algebraic semantics and Jónsson-Tarski representation
AU  - Stefano Bonzio
AU  - Gandolfo Vergottini
AU  - Nicolò Zamperlin
PY  - 2026
DA  - 2026/11/06
PB  - arXiv
DO  - 10.48550/arxiv.2611.09566
UR  - https://arxiv.org/abs/2611.09566
AB  - In this paper we continue the work started in [Bonzio & Zamperlin 2025] by investigating the algebraic semantics for modal external weak Kleene logics. In order to do so we study the global modal logics associated with the three-valued Kripke models introduced in the previous article, showing how these systems are algebraizable. We proceed by defining their algebraic counterparts, obtaining the classes of modal Bochvar and modal Halldén algebras, whose structure theory we investigate via Płonka sum decomposition. Of particular interested is the interaction of the modal operator with the homorphisms of the Płonka decomposition. These results allow us to prove representation theorems for these expanded modal algebras in the style of the Jónsson-Tarski representation for Boolean algebras with operators. As a consequence, we also obtain algebraic semnatics for the local modal logics from which the work originated.
ER  -