In this paper, the notion of direct sum of branches in hks is introduced and some related properties are investigated. Applying this notion to lower hyper $BCK$-semi lattice, a necessary condition for a hi to be prime is given. Some properties of hkc are studied. It is proved that if $H$ is a hkc and $[a)$ is finite for any $ain H$, then $mid Aut(H)mid=1$.

The notion of a (closed) double-framed soft ideal (briefly, a (closed) DFS-ideal) in BCK/BCI-algebras is introduced, and related properties are investigated. Several examples are provided. The relation between a DFS-algebra and a DFS-ideal is considered, and characterizations of a (closed) DFS-ideal are established. A new DFS-ideal from old one is constructed, and we show that the int-uni DFS-s...

We consider the fuzzification of the notion of an n-fold positive implicative ideal. We give characterizations of an n-fold fuzzy positive implicative ideal. We establish the extension property for n-fold fuzzy positive implicative ideals, and state a characterization of PIn-Noetherian BCK-algebras. Finally we study the normalization of n-fold fuzzy positive implicative ideals. 2000 Mathematics...

Abstract We explored new algebra structures such as ? —algebra, —subalgebra and —ideal in this work. In addition, we study their fundamental characteristics connections with other algebras, d/?/BCK - – subalgebras ideals. Also, some examples to illustrate our notions are given. MSC (2010): 20G05, 20D06, 20B30, 20B35