Implicative pre-hilbert algebras and their connections with other algebras of logic

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Document Type : Research Paper

Author

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

Abstract

In the paper, implicative pre-Hilbert algebras are introduced and studied, their characterizations and connections with some algebras of logic are presented. Some important results and examples are given. In particular, it is proven that an implicative pre-Hilbert algebra is equivalent to an implicative pre-BCC algebra. It is shown that for any Hilbert algebra, the implicative property is equivalent to the commutative property. Moreover, several old or new characterizations of Tarski algebras are established. We prove that Tarski algebras coincide with commutative pre-Hilbert algebras and with implicative generalized exchange algebras satisfying the property of antisymmetry. Finally, the hierarchies existing between all classes of implicative algebras considered here are shown.

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References

[1] R. K. Bandaru, A. Borumand Saeid and Y. B. Jun, On GE-algebras, Bull. Sect. Logic, 50 (2021) 81-96.
[2] A. Borumand Saeid, H. S. Kim and A. Rezaei, On BI-algebras, Analele Stiint. ale Univ. Ovidius Constanta Ser. Mat., 25 (2017) 177-194.
[3] A. Borumand Saeid, A. Rezaei, R. K. Bandaru and Y. B. Jun, Voluntary GE-filters and further results of GE-filters in GE-algebras, J. Algebraic Syst., 10 (2022) 31-47.
[4] D. Busneag and S. Rudeanu, A glimpse of deductive systems in algebra, Cent. Eur. J. Math., 8 (2010) 688-705.
[5] A. Diego, Sur les algébras de Hilbert, Collection de Logigue Mathématique, Ser. A, No 21, Gauthier-Villars, Paris, (1966).
[6] L. Henkin, An algebraic characterization of quantifilers, Fundam. Math., 37 (1950) 63-74.
[7] A. Iorgulescu, New generalizations of BCI, BCK and Hilbert algebras-Part I, II, J. Mult.-Valued Log. Soft Comput., 27 (2016) 353-456.
[8] K. Iséki, An algebra related with a propositional calculus, Proc. Jpn. Acad., 42 (1966) 26-29.
[9] K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon., 23 (1978) 1-26.
[10] Y. B. Jun, M. S. Kang, Fuzzifications of generalized Tarski filters in Tarski algebras, Comput. Math. with Appl., 61 (2011) 1-7.
[11] H. S. Kim, Y. H. Kim, On BE-algebras, Scient. Math. Japon., 66 (2007) 113-116.
[12] Y. Komori, The class of BCC-algebras is not a variety, Math. Japon., 29 (1984) 391-394.
[13] C. A. Meredith, Formal Logics, 2nd ed, Oxford, 1962.
[14] A. Monteiro, Lectures on Hilbert and Tarski algebras, Insitituto de Mathemática, Universuidad Nacional del Sur, Bahía Blanca, Argentina, 1960.
[15] S. Tanaka, A new class of algebras, Math. Seminar Notes, 3 (1975) 37-43.
[16] A. Walendziak, The implicative property for some generalizations of BCK algebras, J. Mult.-Valued Log. Soft Comput., 31 (2018) 591-611.
[17] A. Walendziak, The property of commutativity for some generalizations of BCK algebras, Soft Comput., 23 (2019) 7505-7511.
[18] A. Walendziak, On implicative BE algebras, Ann. Univ. Mariae Curie-Skłodowska Lub., Sec. A., 76 (2022) 45-54.
[19] A. Walendziak, On implicative and positive implicative GE algebras, Bull. Sect. Logic, 52 (2023) 497-515.
[20] A. Walendziak, On pre-Hilbert and positive implicative pre-Hilbert algebras, Bull. Sect. Logic, 53 (2024) 345-364.
[21] A. Walendziak, Connections between GE algebras and pre-Hilbert algebras, Alg. Struc. Appl., 11 (2024) 229-241.
[22] Z. Zelent, Transitivity of implicative aBE algebras, Ann. Univ. Mariae Curie-Skłodowska Lub., Sec. A., 76 (2022) 55-58.
Algebraic Structures and Their Applications

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Available Online from 15 April 2025

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