Some connections between BCK-algebras and n-ary block codes

Y. Imai and K. Iseki introduced BCK-algebras in 1966, through the paper [Im, Is; 66], as a generalization of the concept of set-theoretic difference and propositional calculi. This class of BCK-algebras is a proper subclass of the class of BCI-algebras and has many applications to various domains of mathematics. One of the recent applications of BCK-algebras was given in the Coding Theory. In the paper [Ju,So; 11], the authors constructed a finite binary blockcodes associated to a finite BCK-algebra. In [Fl; 15], the author proved that, in some circumstances, the converse of the above statement is also true and in the paper [B,F; 15] the authors proved that binary block codes are an important tool in providing orders with which we can build algebras with some asked properties. For other details regarding BCK-algebras, the reader is referred to [Is, Ta; 78]. In general, the alphabet on which are defined block codes are not binary. It is used an alphabet with n elements, n ≥ 2, identified usually with the set An = {0, 1, 2, ..., n−1}.These codes are called n−ary block codes. In the present paper, we will generalize this construction of binary block codes to n−ary block codes. For this purpose, we will prove that to each n−ary block code V we can associate a BCK-algebra X such that the n−ary block-code generated by X,VX , contains the code V as a subset.