On Pre-Hilbert and Positive Implicative Pre-Hilbert Algebras

Andrzej Walendziak

doi:10.18778/0138-0680.2024.07

Authors

Andrzej Walendziak University of Siedlce, Faculty of Exact and Natural Sciences, Institute of Mathematics https://orcid.org/0000-0002-9866-0781

DOI:

https://doi.org/10.18778/0138-0680.2024.07

Keywords:

Hilbert algebra, pre-Hilbert algebra, BCK-algebra, BCC-algebra, BE-algebra, positive implicativity

Abstract

In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin’s Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.

References

J. C. Abbott, Semi-boolean algebra, Matematički Vesnik, vol. 4(19) (1967), pp. 177–198, URL: https://eudml.org/doc/258960
Google Scholar

R. Bandaru, A. B. Saeid, Y. B. Jun, On GE-algebras, Bulletin of the Section of Logic, vol. 50(1) (2021), pp. 81–96, DOI: https://doi.org/0138-0680.2020.20
Google Scholar

D. Bu¸sneag, S. Rudeanu, A glimpse of deductive systems in algebra, Central European Journal of Mathematics, vol. 8(4) (2010), pp. 688–705, DOI: https://doi.org/10.2478/s11533-010-0041-4
Google Scholar

A. Diego, Sur les algébras de Hilbert, vol. 21 of Collection de Logigue Mathématique, Serie A, Gauthier-Villars, Paris (1966).
Google Scholar

L. Henkin, An algebraic characterization of quantifilers, Fundamenta Mathematicae, vol. 37(1) (1950), pp. 63–74, URL: http://eudml.org/doc/ 213228
Google Scholar

A. Iorgulescu, New generalizations of BCI, BCK and Hilbert algebras— Part I, II, Journal of Multiple-Valued Logic and Soft Computing, vol. 27(4) (2016), pp. 353–456.
Google Scholar

K. Iséki, An algebra related with a propositional calculus, Proceedings of the Japan Academy, vol. 42(1) (1966), pp. 26–29, DOI: https://doi.org/10.3792/pja/1195522171
Google Scholar

K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica, vol. 23(1) (1978), pp. 1–26.
Google Scholar

Y. B. Jun, M. S. Kang, Fuzzifications of generalized Tarski filters in Tarski algebras, Computers and Mathematics with Applications, vol. 61(1) (2011), pp. 1–7, DOI: https://doi.org/10.1016/j.camwa.2010.10.024
Google Scholar

H. S. Kim, Y. H. Kim, On BE-algebras, Scientiae Mathematicae Japonicae, vol. 66(1) (2007), pp. 113–116, DOI: https://doi.org/10.32219/isms.66.1_113
Google Scholar

J. Kim, Y. Kim, E. H. Roh, A note on GT-algebras, The Pure and Applied Mathematics, vol. 16(1) (2009), pp. 59–69, URL: https://koreascience.kr/article/JAKO200910335351650.pdf
Google Scholar

Y. Komori, The class of BCC-algebras is not a variety, Mathematica Japonica, vol. 29 (1984), pp. 391–394.
Google Scholar

C. A. Meredith, Formal logics, 2nd ed., Oxford University Press, Oxford (1962).
Google Scholar

A. Monteiro, Lectures on Hilbert and Tarski algebras, Insitituto de Mathemática, Universuidad Nacional del Sur, Bahía Blanca, Argentina (1960).
Google Scholar

A. Walendziak, On commutative BE-algebras, Scientiae Mathematicae Japonicae, vol. 69(2) (2009), pp. 281–284, DOI: https://doi.org/10.32219/isms.69.2_281
Google Scholar