Complete and atomic Tarski algebras

  • Sergio Arturo CelaniEmail author


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.


Tarski algebras Tarski sets Representation theorem Complete and atomic Tarski algebras Modal operator 

Mathematics Subject Classification

03B45 03G25 


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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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemática, Facultad de Ciencias ExactasUniversidad Nacional del Centro de la Provincia de Buenos AiresTandilArgentina

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