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Complete and atomic Tarski algebras

  • Sergio Arturo CelaniEmail author
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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.

Keywords

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

Mathematics Subject Classification

03B45 03G25 

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Notes

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).

References

  1. 1.
    Abad, M., Dias Varela, J.P., Zander, M.: Varieties and quasivarieties of monadic tarski algebras. Sientiae Math. Jpn. 56(3), 599–612 (2002)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abbott, J.C.: Semi-boolean algebras. Mater. Vesn. 4(19), 177–198 (1967)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abbott, J.C.: Implicational algebras. Bull. Math. R. Soc. Roum. 11, 3–23 (1967)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Busneag, D.: On the maximal deductive systems of a bounded Hilbert algebra. Bull. Math. Soc. Sci. Math. Roum. Tomo 31(79), 1–13 (1987)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Celani, S.A.: A note on homomorphism of Hilbert algebras. Int. J. Math. Math. Sci. 29(1), 55–61 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Celani, S.A.: Modal tarski algebras. Rep. Math. Log. 39, 113–126 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chajda, I., Halaš, P., Zedník, J.: Filters and annihilators in implication algebras. Acta Univ. Palacki. Olomuc. Fac. Rer. Nat. Math. 37, 41–45 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Diego A.: Sur les algébras de Hilbert. Colléction de Logique Mathèmatique, serie A, 21, Gouthier-Villars, Paris (1966)Google Scholar
  9. 9.
    Givant, S.: Duality theories for Boolean Algebras with Operators. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Givant, S., Halmos, P.: Introduction to Boolean Algebras, Undergraduate Texts in Mathematics. Springer, New York (2009)zbMATHGoogle Scholar
  11. 11.
    Jarvinen, J.: On the structure of rough approximations. Fund. Inf. 53, 135–153 (2002)MathSciNetzbMATHGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Monteiro A.: Sur les algèbres de Heyting symétriques. Portugaliae Mathematica 39, fasc. 1–4 (1980)Google Scholar
  14. 14.
    Thomason, S.K.: Categories of frames for modal logic. J. Symb. Log. 40, 439–442 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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