Block code on L-algebras

Document Type : Original Article

Author

Department of Mathematics, Shahid Beheshti University, Tehran, Iran

Abstract

By using the notion of L-algebras as an important part of the ordered algebra, we introduce the notions of block code, x-function and x-subsets on an arbitrary L-algebra. Then some related properties and examples are provided. Also, by  using these notions, we define an equivalence relation on L-algebra and we introduce a new order on the generated code based on L-algebras. Finally, we will provide a method which allows us to find an L-algebra starting from a given arbitrary binary block code. 

Keywords


[1] M. Afshar Jahanshahi, A. Borumand Saeid, Binary block codes of MV -algebras and Fibonacci sequences, Algebraic Structures and Their Applications, 2022, to appear.
[2] H. Bordbar, The structure of the block code generated by a BL-algebra, Mathematics, 10 (2022), 692.
[3] B. Bosbach, Concerning cone algebras, Algebra Universalis, 15 (1982), 58–66.
[4] C. Dietzel, P. Menchómn, L. Vendramis, On the enumeration of finite L-algebra, arXive: 2206.04955v1, 10 June 2022.
[5] V. G. Drinfeld, On some unsolved problems in quantum group theory, in: P.P. Kulish (Ed.), Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, Vol. 1510, Springer, Berlin, 1992.
[6] P. Etingof, Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Math, (2001).
[7] P. Etingof, S. Gelaki, A method of construction of finite-dimensional triangular semisimple Hopf algebras, Mathematical Research Letters, 5 (1998), 551–561.
[8] P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Mathematical Journal, 100 (1999), 169–209.
[9] C. Flaut, BCK-algebras arising from block codes, Journal of Intelligent and Fuzzy Systems, 28 (2015), 1829-1833. DOI:10.3233/IFS-141469.
[10] T. Gateva-Ivanova, Noetherian properties of skew-polynomial rings with binomial relations, Transactions of the American Mathematical Society, 343 (1994), 203–219.
[11] T. Gateva-Ivanova, Skew polynomial rings with binomial relations, Journal of Algebra, 185 (1996), 710–753.
[12] T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I-type, Journal of Algebra, 206 (1998), 97–112.
[13] Y.B. Jun, A.Z. Song, Codes based on BCK-algebras, Information Science, 181 (2011), 5102- 5109.
[14] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Advances in Mathematics, 193 (2005), 40–55.
[15] W. Rump, L-algebras, self-similarity, and l-groups, Journal of Algebra, 320 (2008), 2328–2348.
[16] W. Rump, Semidirect products in algebraic logic and solutions of the quantum Yang-Baxter equation, Journal of Algebra and Applications, 7 (2008), 471–490.
[17] W. Rump, Y. Yang, Interval in ℓ-groups as L-algebras, Algebra Universalis, 67(2) (2012), 121–130.
[18] J. Tate, M. Van den Bergh, Homological properties of Sklyanin algebras, Inventiones Mathematicae, 124 (1996), 619–647.
[19] T. Traczyk, On the structure of BCK-algebras with zxyx = zyxy, Japanese Journal of Math[1]ematics, 33 (1988), 319–324.
[20] A. P. Veselov, Yang-Baxter maps and integral dynamics, Physics Letters A, 314(3) (2002), DOI:10.1016/S0375-9601(03)00915-0.