EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 2, 2018, 444-448 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Lie algebras with BCL algebras Yonghong Liu School of Automation, Wuhan University of Technology, Wuhan 430070, China Abstract. The subject matter of this work is hoping for a new relationship between the Lie algebras and the algebra of logic, which will constitute an important part of our study of “pure” algebra theory. BCL algebras as a class of logical algebras can be generated by a Lie algebra. The opposite is also true that when special conditions occur. The aim of this paper is to prove several theorems on Lie algebras with BCL algebras. I introduce the notion of a “pseudo-association” which I propose as the adjoint notion of BCL algebra in the abelian group. 2010 Mathematics Subject Classifications: 17B60, 03G25 Key Words and Phrases: Lie algebras, BCL algebras, abelian Lie algebras, pseudo-association BCL algrbra 1. Introduction Lie algebras comprise a significant part of Lie group theory (see [1]) and are being vibrantly studied. On the other hand, Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. But it is sig- nificant that our results show that the Lie algebra and logical algebra are closely linked. Sure, BCL algebras as a class of logical algebras were introduced by Liu in 2011 [2]. The last results was discovered and developed in [3-15]. From set theory perspective, BCL algebras are the algebraic formulations of the set difference together with its properties. In the paper, I just want to prove that the connectivity theorems but that I have sus- pected for a long time, which is the relationship between the Lie algebras and the BCL algebras. More importantly, we developed the theory that Lie algebras do have a pre- ferred direction that causes us to the study of logic issues so we can capture new method. Meanwhile, let the theory of BCL algebras becomes strong enough. Email address: hylinin@163.com (Y. Liu) http://www.ejpam.com 444 c© 2018 EJPAM All rights reserved. Yonghong L. / Eur. J. Pure Appl. Math, 11 (2) (2018), 444-448 445 2. Basic definitions In this section, we list two definitions from the literature that will be used in the sequel. Definition 2.1 A Lie algebra over a field k is a vector space g over k together with a k-bilinear map [ , ]: g × g → g (called the bracket) such that (Lie 1) [x, x] = 0 for all x ∈ g (Lie 2) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ g. A homomorphism of Lie algebras is a k-linear map α: g → g′ such that α([x, y]) = [α(x), α(y)] for all x, y, z ∈ g. Condition (Lie 2) is called the Jacobi identity. Note that (Lie 1) applied to [x + y, x + y] shows that the Lie bracket is skew-symmetric. [x, y] = − [y, x] for all x, y, z ∈ g. Definition 2.2 ([2], Definition 2.1.) A BCL algebra is a triple (A; →, 0), where A is a nonempty set, → is a binary operation on A, the following three axioms hold for any x, y, z ∈ A. (BCL 1) x → x = 0. (BCL 2) x → y = 0 and y → x = 0 imply x = y. (BCL 3) (((x → y) → z) → ((x → z) → y)) → ((z → y) → x) = 0. 3. Results Theorem 3.1 Let L be Lie algebras. Define x → y = [x, y]− [y, x], (x → y) → z = [[x, y], z], and 0 → x = 0 = [x, x]. Then L be BCL algebras. Proof. Let x, y, z ∈ L. Then Yonghong L. / Eur. J. Pure Appl. Math, 11 (2) (2018), 444-448 446 (1) x → x = [x, x]− [x, x] = 0. (2) [x, x] = 0 = x → y and [y, y] = 0 = y → x imply x = y. (3) [[[[x, y], z], [[x, z], y]], [[z, y], x]] = 0. Clearly, proving (1) and (2). Now we need to prove (3), we define [x, y] = x + y. Then [[z, y], x] = [x, [z, y]] ⊆ [z, [y, x]] + [y, [x, z]] = [[x, y], z] + [[x, z], y] ⊆ [[x, y], z], [[x, z], y]], and (3) is proved. We see that L be BCL algebras. Theorem 3.2 Let x, y, z ∈ P be BCL algrbras. Then P is abelian Lie algebra iff x = y = z. Proof. Assume that P is abelian Lie algebra, sine x, y, z ∈ P , we have [x, y] = 0 = [y, z]. Therefore, x = y = z. Conversely, assume x = y = z. To prove that this algebra is a BCL algebra. Let x, y, z ∈ P . By Theorem 2.1. Then (4) [x, x] = 0 = x → x. (5) [x, x] = 0 = x → y and [y, y] = 0 = y → x imply x = y. (6) [0, [0, x]] = [[0, x], 0] = (0 → x)→ 0 = 0→ 0 = 0. This completes the proof. Definition 3.1 Let (G, +) be an abelian, (G; −, 0) be an adjoint BCL algrbras and Yonghong L. / Eur. J. Pure Appl. Math, 11 (2) (2018), 444-448 447 (G; →, 0) be a pseudo-association BCL algrbras. Suppose the following conditions hold: (GPA 1) x − (0 − y) = x + y. (GPA 2) x − y = x→ y. Then adjoint group of (G; −, 0) is abelian (G, +), and adjoint BCL algrbras of abelian (G, +) is (G; →, 0). Theorem 3.3 Let P be a pseudo-association BCL algrbra. The bracket [x, y] = x → y − (y → x), for all x, y ∈ P . Then P be a Lie algebras about the bracket [ , ] and we use notation PL, for the Lie algebra is generated by the pseudo-association BCL algrbra. Proof. By definition of the bracket, [x, x] = 0 trivially hoid. To prove bilinear, sine x1, x2, y ∈ PL, and λ1, λ2 ∈ PL, we have [λ1 x1 + λ2 x2, y] = ((λ1 x1 + λ2 x2)→ y)− (y → (λ1 x1 + λ2 x2)) = (λ1(x1 → y) + λ2(x2 → y))− (λ1(y → x1) + λ2(y → λ2 x2)) = λ1[x1, y] + λ2[x2, y] To prove Jacobi identity, sine x, y, z ∈ PL, we have (7) [x, [y, z]] = [x, y → z − (z → y)] = x→ (y → z − (z → y))− ((y → z − (z → y))→ x) = x→ y → z − x→ z → y − y → z → x+ z → y → x. (8) [y, [z, x]] = y → z → x− y → x→ z − z → x→ y + x→ z → y. (9) [z, [x, y]] = z → x→ y − z → y → x− x→ y → z + y → x→ z. Therefore, the sum of three brackets, i.e., (7), (8) and (9) satisfying the Jacobi iden- tity (10) [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. REFERENCES 448 References [1] Serre J-P. Lie Algebras and Lie Groups (2nd ed.), Springer (2006) [2] Yonghong L. A new branch of the pure algebra: BCL-algebras, Advances in Pure Mathematics, 1, 297-299 (2011), https://dx.doi.org/10.4236/apm.2011.15054 [3] Yonghong L. On BCL+-algebras, Advances in Pure Mathematics, 2, 59-61 (2012), https://dx.doi.org/10.4236/apm.2012.21012 [4] Al-Kadi D. and Hosny R. On BCL-algebra, Journal of Advances in Mathematics, 3, 184-190 (2013). [5] Al-Kadi D. Soft BCL-algebra, International Journal of Algebra, 8, 57-65 (2014), https://dx.doi.org/10.12988/ija.2014.311122 [6] Yonghong L. Partial orders in BCL+-algebra, Journal of Advances in Mathematics, 5, 630-634 (2013). [7] Yonghong L. Topological BCL+-algebras, Pure and Applied Mathematics Journal, 3, 11-13 (2014). [8] Yonghong L. Some distributions of BCL+-algebras, International Journal of Algebra, 8, 495-503 (2014). https://dx.doi.org/10.12988/ija.2014.4556 [9] Yonghong L. Filtrations and deductive systems in BCL+ algebras, British Journal of Mathematics & Computer Science, 8, 274-285 (2015). [10] Yonghong L. Funnels in BCL+ algebras, International Journal of Mathematical Sciences & Engineering Applications, 9, 179-185 (2015). [11] Yonghong L. Liu’s laws and p-BCL+ algebras, International Journal of Pure & Engineering Mathematics, 3, 51-58 (2015). [12] Yonghong L. Standard ideals in BCL+ algebras, Journal of Mathematics Research, 8, 37-44 (2016). [13] Yonghong L. On Liu algebras: a new composite structure of the BCL+ algebras and the semigroups, Journal of Semigroup Theory and Applications, 2, 1-16 (2017). [14] Khalil S. M. and Musawi A. F. M. J. Soft BCL-algebras of the power sets, International Journal of Algebra, 11, 329-341 (2017), https://doi.org/10.12988/ija.2017.7735 [15] Yonghong L. pa-BCL+ algebras and groups, Applied Mathematics & Information Sciences, 11, 891-897 (2017), http://dx.doi.org/10.18576/amis/110329