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

The purpose of this document is to develop some the basic theory multipliers algebra dual BCK-algebras. In study we demonstrate concept left bi-multiplier and right BCK (DBCK-algebra) algebras. Several examples results pertaining these on DBCK-algebras are developed based definitions. Then characteristics bi-multipliers obtained properties DBCK-algebras. We focused behavior elements under also ...

Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of BCK-algebras. The notions of positive implicative soft ideals and positive implicative idealistic soft BCK-algebras are intro...

In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of double-framed soft sets is introduced, and applications in BCK/BCI-algebras are discussed. The notions of double-framed soft algebras in BCK/BCIalgebras are introduced, and related properties are investigated. Characterizations of doubleframed soft algebras are...

We explore properties of the set of d-units of a d-algebra. A property of interest in the study of d-units in d-algebras is the weak associative property. It is noted that many other d-algebras, especially BCK-algebras, are in fact weakly associative. The existence of d/BCK-algebras which are not weakly associative is demonstrated. Moreover, the notions of a d-integral domain and a left-injecti...

It is well-known that the representation of several classes of residuated lattices involves lattice-ordered groups. An often applicable method to determine the representing group (or groups) from a residuated lattice is based on partial algebras: the monoidal operation is restricted to those pairs which fulfil a certain extremality condition, and else left undefined. The subsequent construction...

Starting from quasi-Wajsberg algebras (which are generalizations of Wajsberg algebras), whose regular sets are Wajsberg algebras, we introduce a theory of quasi-algebras versus, in parallel, a theory of regular algebras. We introduce the quasi-RM, quasi-RML, quasi-BCI, (commutative, positive implicative, quasi-implicative, with product) quasi-BCK, quasi-Hilbert and quasi-Boolean algebras as gen...

For any σ-complete Boolean algebras A and B, if A is isomorphic to [0, b] ⊆ B and B is isomorphic to [0, a] ⊆ A, then A B. Recently, several generalizations of this known CantorBernstein type theorem for MV-algebras, (pseudo) effect algebras and `-groups have appeared in [1], [2], [4] and [5]. We prove an analogous result for certain pseudo BCK-algebras—a noncommutative extension of BCK-algebra...

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...