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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References
Abad, M., Dias Varela, J.P., Zander, M.: Varieties and quasivarieties of monadic tarski algebras. Sientiae Math. Jpn. 56(3), 599–612 (2002)
Abbott, J.C.: Semi-boolean algebras. Mater. Vesn. 4(19), 177–198 (1967)
Abbott, J.C.: Implicational algebras. Bull. Math. R. Soc. Roum. 11, 3–23 (1967)
Busneag, D.: On the maximal deductive systems of a bounded Hilbert algebra. Bull. Math. Soc. Sci. Math. Roum. Tomo 31(79), 1–13 (1987)
Celani, S.A.: A note on homomorphism of Hilbert algebras. Int. J. Math. Math. Sci. 29(1), 55–61 (2002)
Celani, S.A.: Modal tarski algebras. Rep. Math. Log. 39, 113–126 (2005)
Chajda, I., Halaš, P., Zedník, J.: Filters and annihilators in implication algebras. Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Math. 37, 41–45 (1998)
Diego A.: Sur les algébras de Hilbert. Colléction de Logique Mathèmatique, serie A, 21, Gouthier-Villars, Paris (1966)
Givant, S.: Duality theories for Boolean Algebras with Operators. Springer, Berlin (2014)
Givant, S., Halmos, P.: Introduction to Boolean Algebras, Undergraduate Texts in Mathematics. Springer, New York (2009)
Jarvinen, J.: On the structure of rough approximations. Fund. Inf. 53, 135–153 (2002)
Kondo M.: Algebraic approach to generalized rough sets, In: Wang, D., Szczuka, G.M., Düntsch, I., Yao, Y. (eds.) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC: Lecture Notes in Computer Science, vol. 3641, p. 2005. Springer, Berlin (2005)
Monteiro A.: Sur les algèbres de Heyting symétriques. Portugaliae Mathematica 39, fasc. 1–4 (1980)
Thomason, S.K.: Categories of frames for modal logic. J. Symb. Log. 40, 439–442 (1975)
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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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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DOI: https://doi.org/10.1007/s00153-019-00666-x