Some results on commutative BI-Algebras‎

Document Type : Research Paper

Authors

Department of Mathematics, Payame Noor University, p.o.box. 19395-3697, Tehran, Iran

Abstract

‎The notion of a (branchwise) commutative $BI$-algebra is presented, and some related properties are investigated. We show that the class of commutative $BH$-algebras is broader than the class of commutative $BI$-algebras. Moreover, we %show that prove every  singular $BI$-algebra is a $BH$-algebra. Also, we define the  commutative ideals in $BI$-algebras and characterize the commutative $BI$-algebras in terms of commutative ideals.

Keywords

Main Subjects

References

[1] J. C. Abbott, Semi-Boolean Algebras, Matematicki Vesnik vol. 4 (1967) 177{198.
[2] S. S. Ahn, J. M. Ko, and A. Borumand Saeid, On Ideals of BI-Algebras, Journal of the Indonesian Mathematical Society vol. 25, no. 1 (2019) 24{34.
[3] R. K. Bandaru, On QI-Algebras, Discussiones Mathematicae General Algebra and Applications vol. 37 (2017) 137{145.
[4] A. Borumand Saeid, H. S. Kim, and A. Rezaei, On BI-Algebras, Analele Stiinti ce ale Universitatii \Ovidius" Constanta. Seria Matematica vol. 25 (2017) 177{194.
[5] W. Y. Chen, and J. S. Oliveira, Implication Algebras and the Metropolis Rota Axioms for Cubic Lattices, Journal of Algebra vol. 171 (1995) 383{396.
[6] A. Diego, Sur Algebres de Hilbert, Collection de Logique Mathematique, Series A vol. 21 (1967) 177{198.
[7] R. Halas, Remarks on Commutative Hilbert Algebras, Mathematica Bohemica vol. 127, no. 4 (2002) 525{529.
[8] L. Henkin, An Algebraic Characterization of Quanti ers, Fundamenta Mathematicae vol. 37 (1950) 63{74.
[9] Y. Imai and K. Iseki, On Axioms Systems of Propositional Calculi XIV, Proceedings of the Japan Academy, Series A vol. 42 (1966) 19{22.
[10] A. Iorgulescu, New Generalizations of BCI, BCK and Hilbert Algebras { Part I, Journal of Multiple-Valued Logic and Soft Computing vol. 27 (2016) 353{406.
[11] A. Iorgulescu, New Generalizations of BCI, BCK and Hilbert Algebras { Part II, Journal of Multiple-Valued Logic and Soft Computing vol. 27 (2016) 407{456.
[12] Y. B. Jun, Commutative Hilbert Algebras, Soochow Journal of Mathematics vol. 22 (1996) 477{484.
[13] Y. B. Jun, E. H. Roh, and H. S. Kim, On BH-Algebras, Scientiae Mathematicae Japonicae vol. 1 (1998) 347{354.
[14] H. S. Li, An Axiom System of BCI-Algebras, Mathematica Japonica vol. 30 (1985) 351{352.
[15] J. Meng, Implication Commutative Semigroups are Equivalent to a Class of BCK-Algebras, Semigroup Forum vol. 50 (1995) 89{96.
[16] J. Meng, Implication Algebras are Dual to Implicative BCK-Algebras, Soochow Journal of Mathematics vol. 22, no. 4 (1996) 567{571.
[17] S. Niazian, On hyper BI-algebras, Journal of Algebraic Hyper Structures and Logical Algebras vol. 2, no. 1 (2021), 47{67.
[18] A. Radfar, Classi cation BI-algebras of order less than 5, submitted.
[19] A. Rezaei and F. Smarandache, The Neutrosophic Triplet of BI-algebras Neutrosophic Sets and Systems vol. 33 (2020) 313{321.
[20] A. Rezaei and S. Soleymani, Applications of states to BI-algebras, Journal of Algebraic Hyper Structures and Logical Algebras vol. 3, no. 3 (2022), 45{63.
[21] S. Tanaka, A New Class of Algebras, Mathematics Seminar Notes vol. 3 (1975) 37{43.
[22] S. Tanaka, On ^-Commutative Algebras, Mathematics Seminar Notes vol. 3 (1975) 59{64.
[23] S. Tanaka, On ^-Commutative Algebras II, Mathematics Seminar Notes vol. 5 (1975) 245{247.
[24] A. Walenziak, The Property of Commutativity for Some Generalizations of BCK-Algebras, Soft Computing vol. 23, no. 17 (2019) 7505{7511.
[25] H. Yisheng, BCI-Algebra, Science Press, Beijing, 2006.
Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 491-510

Files

History

  • Receive Date: 30 March 2023
  • Revise Date: 19 July 2023
  • Accept Date: 06 October 2023

Share

How to cite

Statistics