New elements in Hilbert algebras
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematics, Behbahan Branch, Islamic Azad University, Behbahan, Iran.
Keywords: Hilbert algebras, commutator, Engel (involution) elements, left (right) $k$-Engel element.,
Abstract :
In this paper, the notion of commutator of elements of a Hilbert algebra are introduced and some properties are given. The notions of involution element and Engel element in Hilbert algebras are introduced. Many different characterizations of them are given. Then, left (right) $k$-Engel elements as a natural generalization of commutators are introduced, and we discuss Engel elements, which is defined by left and right commutators. Finally, we will also study relationships between these elements.
[1] Diego A. Sur algbres de Hilbert. Collect. Logique Math. Ser. A. 1967; 21: 177-198.
[2] Busneag D. A note on deductive systems of a Hilbert algebra. Kobe journal of mathematics. 1985; 2: 20-35.
[3] Jun YB, Park CH. Applications of soft sets in Hilbert algebras. Iranian Journal of Fuzzy Systems. 2009; 6(2): 7586. DOI: http://doi.org/10.22111/ijfs.2009.212
[4] Chajda I, Hala R. Semilattice Structures. Heldermann Verlag: Research and Exposition in Mathematics; 2001.
[5] Jun YB, Nam JW, Hong SM. A note on Hilbert algebra. East Asian mathematical journal. 1994; 10(2): 279-285. DOI: https://doi.org/10.1142/S1793557109000303
[6] Najafi A, Borumand Saeid A. Engel BCI-algebras: An application of left and right commutators. Mathematica Bohemica. 2021; 146(2): 133-150. DOI: http://doi.org/10.21136/MB.2020.0160-18
[7] Chajda I, Hala R. Congruences and ideals in Hilbert algebras. Kyungpook Mathematical Journal. 1999; 2: 429-432. DOI: https://doi.org/10.1515/dema-2000-0202
[8] Sergio A. Notes on bounded Hilbert algebras with supremum. Acta Scientiarum Mathematicarum. 2014; 80(1): 3-19. DOI: https://doi.org/10.14232/actasm-012-267-9
