The Structure of the Block Code Generated by a BL-Algebra

Abstract

Inspired by the concept of BL-algebra as an important part of the ordered algebra, in this paper we investigate the binary block code generated by an arbitrary BL-algebra and study related properties. For this goal, we initiate the study of the BL-function on a nonempty set P based on BL-algebra L, and by using that, l-functions and l-subsets are introduced for the arbitrary element l of a BL-algebra. In addition, by the mean of the l-functions and l-subsets, an equivalence relation on the BL-algebra L is introduced, and using that, the structure of the code generated by an arbitrary BL-algebra is considered. Some related properties (such as the length and the linearity) of the generated code and examples are provided. Moreover, as the main result, we define a new order on the generated code C based on the BL-algebra L, and show that the structures of the BL-algebra with its order and the correspondence generated code with the defined order are the same.

A Binary Block Code Generated by JU-Algebras

“…In this study, the method used in generating binary block code in BL-algebras, KU -algebras, and BCKalgebras will be applied to JU -algebras. Relationships between the generated code and JU -algebras are obtained analogous to the results found in [3], [8]- [11].…”

“…In this article, the binary block code generated by JU -algebra were studied. Results parallel to [8], [3], [9] were generated. It is worth noting that the JU -code is not linear.…”

Preface to the Special Issue “Algebraic Structures and Graph Theory”

“…Based on the notion of the Hopf Galois extension, the author emphasizes the relationships between the derived categories D(R#H) of the smash product R#H and D(R H ), where H is a finite dimensional semisimple Hopf algebra and R a left H-module algebra. The second paper [9] studies the structure of a block code generated by a BL-algebra. In particular, H. Bordbar defines a new order for the generated code associated with a BL-algebra, and shows that the structure of the BL-algebra with its initial order and the one of the corresponding generated code with the new defined order coincide.…”

Codes based on basic algebras

“…Jun and Song applied coding theory to these algebras in [14]. Recently, binary block codes produced by different ordered algebraic structures have been investigated by many mathematicians in [1], [2], [10], [11], [12], [15], [16], [17].…”