Connections between GE algebras and pre-Hilbert algebras

Document Type : Research Paper

Author

Institute of Mathematics, Faculty of Exact and Natural Science, Siedlce University, PL-08110 Siedlce, Poland

Abstract

GE algebras (generalized exchange algebras) and pre-Hilbert algebras are a generalization of well-known Hilbert algebras. In the paper, connections between these algebras are studied. In particular, it is proven that pi-BE algebras are equivalent with GE algebras satisfying the exchange property. Some characterizations of transitive GE algebras and exchange pre-Hilbert algebras are given. It is shown that the intersection of classes of GE algebras and pre-Hilbert algebras is the class of transitive GE algebras. Moreover, GE, BE and pre-Hilbert algebras with the antisymmetry property are investigated. It is proven that transitive GE algebras satisfying the property of antisymmetry coincide with Hilbert algebras. Finally, positive implicative GE and pre-Hilbert algebras are considered, their connections with some algebras of logic are presented. In addition, the hiearchies existing between the classes of algebras studied here are shown.

Keywords

References

[1] J. C. Abbott, Semi-boolean algebras, Mat. Vesnik, 4 No. 19 (1967) 177-198.
[2] R. K. Bandaru, A. Borumand Saeid and Y. B. Jun, On GE-algebras, Bull. Sect. Logic, 50 No. 1 (2021) 81-96.
[3] R. K. Bandaru, A. Borumand Saeid and Y. B. Jun, Belligerent GE-filters in GE-algebras, J. Indones. Math. Soc., 28 (2021) 31-43.
[4] A. Borumand Saeid, A. Rezaei, R. K. Bandaru and Y. B. Jun, Voluntary GE-filters and futher results of GE-filters in GE-algebras, J. Algebr. Syst., 10 No. 1 (2022) 31-47.
[5] D. Busneag and S. Rudeanu, A glimpse of deductive systems in algebra, Cent. Eur. J. Math., 8 No. 4 (2010) 688-705.
[6] P. Cintula and C. Noguera, Logic and Implication: An Introduction to the General Algebraic Study of Non-Classical Logics, Trends in Logic 57, Springer, Berlin, 2021.
[7] A. Diego, Sur les algébras de Hilbert, Collection de Logigue Mathématique, Edition Hermann, Serie A, XXI, 1966.
[8] L. Henkin, An algebraic characterization of quantifilers, Fundam. Math., 37 (1950) 63-74.
[9] A. Iorgulescu, New generalizations of BCI, BCK and Hilbert algebras - Part I, J. Mult.-Valued Log. Soft Comput., 27 No. 4 (2016) 353-406.
[10] A. Iorgulescu, New generalizations of BCI, BCK and Hilbert algebras - Part II, J. Mult.-Valued Log. Soft Comput., 27 No. 4 (2016) 407-456.
[11] K. Iséki, An algebra related with a propositional calculus, Proc. Jpn. Acad., 42 (1966) 26-29.
[12] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon., 23 (1978) 1-26.
[13] H. S. Kim and Y. H. Kim, On BE-algebras, Scient. Math. Japon., 66 No. 1 (2007) 113-116.
[14] C. A. Meredith, Formal Logics, 2nd ed., Oxford, 1962.
[15] A. Monteiro, Lectures on Hilbert and Tarski Algebras, Insitituto de Mathemática, Universuidad Nacional del Sur, Bahía Blanca, Argentina, 1960.
[16] A. Rezaei, A. Borumand Saeid and R. A. Borzooei, Relation between Hilbert algebras and BE-algebras, Appl. Math., 8 (2013) 573-584.
[17] A. Rezaei, R. K. Bandaru, A. Borumand Saeid and Y. B. Jun, Prominent GE-filters and GE-morphisms in GE-algebras, Afr. Mat., 32 (2021) 1121-1136.
[18] S. Z. Song, R. K. Bandaru and Y. B. Jun, Imploring GE-filters of GE-algebras, J. Math., 2021, 1-7.
[19] S. Z. Song, R. K. Bandaru, D. A. Romano and Y. B. Jun, Interior GE-filters of GE-algebras, Diss. Math. Gen. Alg. App., 42 (2022) 217-235.
[20] A. Walendziak, On commutative BE-algebras, Scient. Math. Japon., 69 No. 2 (2009) 281-284.
[21] A. Walendziak, On implicative and positive implicative GE algebras, Bull. Sect. Logic, 1-19.
[22] A. Walendziak, On pre-Hilbert and positive implicative pre-Hilbert algebras, Submitted.
Algebraic Structures and Their Applications
Volume 11, Issue 3
August 2024
Pages 229-241

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