In this paper, we define the concept of generalized cubic subsemigroups (ideals) of a semigroup and investigate some of its related properties. In particular, we introduce the concept of $(\in _{(\widetilde{\gamma }_{1},\gamma _{2}) },\in _{(\widetil

Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.

The concept of quantum B-algebra was introduced by Rump and Yang, that is, unified algebraic semantics for various noncommutative fuzzy logics, quantum logics, and implication logics. In this paper, a new notion of q-filter in quantum B-algebra is proposed, and quotient structures are constructed by q-filters (in contrast, although the notion of filter in quantum B-algebra has been defined before this paper, but corresponding quotient structures cannot be constructed according to the usual methods). Moreover, a new, more general, implication algebra is proposed, which is called basic implication algebra and can be regarded as a unified frame of general fuzzy logics, including nonassociative fuzzy logics (in contrast, quantum B-algebra is not applied to nonassociative fuzzy logics). The filter theory of basic implication algebras is also established.

As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in $BCK/BCI$ -algebra is considered. The notions of $\alpha$ -internal, $\beta$ -internal, $\alpha$ -external and $\beta$ -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of $\alpha$ -internal and $\alpha$ -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in $BCK/BCI$ -algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed.