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The concept of ₦-structures was initiated by Jun et al. in 2009. Later, Young Bae Jun extended this concept as ₦-hyper sets and dealt its associated outcomes. In particular, four types of ₦-substructures with relevant properties have been investigated which leads to the introduction of the notion of ₦-hyper [Formula: see text]subalgebra in this study. Further, it was proved that the intersection of two ₦-hyper [Formula: see text]subalgebras is again an ₦-hyper [Formula: see text]subalgebra. In addition to that, the properties based on the homomorphism and Cartesian product were discussed.

Recently, a number of researches studied relationship between codes and BCK-algebras, dealing with the category codes in a BCK-algebra have not been considered in earlier works. This paper investigates a code constructed by a BCK-algebra and also a BCK-algebra constructed based on code. The suggested rendered algorithm constructs the code based on BCK-algebra, the fixed dimension of this code is guaranteed by a BCK-algebra with some condition. We introduce the category of linear codes and denoted by [Formula: see text]. In addition, a subcategory of BCK-algebra is described and denoted by [Formula: see text]. Finally, we prove that [Formula: see text] is equivalent to [Formula: see text].

In this paper, we introduce and study the notion of a (positive implicative) left ray in groupoids, and we show that every normal subgroup of a group is a left ray of a group, and in every finite group, left rays are normal subgroups. Further, left absorptive subsets of groupoids are discussed and several examples for these definitions provided. The relationship between left rays and zero semigroups is discussed.

In this paper, we introduce the notion of a wRM/eRM-algebra as a generalization of a RM-algebra. These structures are studied in details. Also, we derived an eRM-algebra from a RM-algebra and vice versa. The concept of a positive implicative eRM-algebra is defined and we discussed on the medial filters.