In this paper, we introduce the notions of Belluce lattice associated with a bounded $BCK$-algebra and reticulation $BCK$-algebra. To do this, first, define operations $curlywedge ,$ $curlyvee $ $sqcup on $BCK$-algebras study some algebraic properties them. Also, for $A$ Zariski topology Spec(A)$ induced $tau _{A,Max(A)}$ $Max(A)$. We prove $(Max(A),tau_{A,Max(A)})$ is compact topological space...

Cantor's naive set theory is characterized by the unrestricted comprehension principle, saying that for every formula A(x), there exists a set {x|A(x)} such that A(t) ↔ t ∈ {x|A(x)} for any term t. The theory is intuitive, elegant, powerful, but unfortunately inconsistent (as witnessed by Russell's paradox). While it is common to somehow restrict the comprehension scheme (as in ZFC), there is a...

A Heyting algebra is a distributive lattice with implication and a dual $BCK$-algebra is an algebraic system having as models logical systems equipped with implication. The aim of this paper is to investigate the relation of Heyting algebras between dual $BCK$-algebras. We define notions of $i$-invariant and $m$-invariant on dual $BCK$-semilattices and prove that a Heyting semilattice is equiva...

This paper deals with the study of generalizations fuzzy subalgebras and ideals in BCK/BCI-algebras. In fact, notions ∈ , ∨ κ ~ ∗ q -fuz...

We consider the intuitionistic fuzzification of the concept of subalgebras and ideals in BCK-algebras, and investigate some of their properties. We introduce the notion of equivalence relations on the family of all intuitionistic fuzzy ideals of a BCK-algebra and investigate some related properties.

The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored.

In this paper by considering the notion of BCK-module X, we introduce new concept which is named extended BCK-module or briefly X-module and we obtain some interesting results by it. Furthermore, we define the notion of multiplication X-module. 2000 Mathematics subject classification: 06D99 • 08A30

first we show that the cosets of a fuzzy ideal μ in a bck-algebrax form another bck-algebra  x/μ (called the fuzzy quotient bck-algebra of x by μ). also we show thatx/μ is a fuzzy partition of x and we prove several some isomorphism theorems. moreover we prove that if the associated fuzzy similarity relation of a fuzzy partition p of a commutative bck-algebra iscompatible, then p is a fuzzy quo...