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Complete and atomic Tarski algebras

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Abstract

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the \(\left\{ \rightarrow \right\} \)-subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures \(\left<X,{\mathcal {K}}\right>\), where X is a non-empty set and \({\mathcal {K}}\) is non-empty family of subsets of X such that \(\bigcup {\mathcal {K}}=X\). This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.

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Acknowledgements

This paper has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 689176, and the support of the Grant PIP 11220150100412CO of CONICET (Argentina).

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  1. Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina

    Sergio Arturo Celani

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Correspondence to Sergio Arturo Celani.

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Celani, S.A. Complete and atomic Tarski algebras. Arch. Math. Logic 58, 899–914 (2019). https://doi.org/10.1007/s00153-019-00666-x

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