###### On BI-Algebras

## Abstract

In this paper, we introduce a new algebra, called a BI-algebra, which is a generalization of a (dual) implication algebra and we discuss the basic properties of BI-algebras, and investigate ideals and congruence relations.

###### An Injective Pseudo-BCI Algebra is Trivial

## Abstract

Injective pseudo-BCI algebras are studied. There is shown that the only injective pseudo-BCI algebra is the trivial one.

###### On QI-Algebras

## Abstract

In this paper, the notion of a QI-algebra is introduced which is a generalization of a BI-algebra and there are studied its properties. We considered ideals, congruence kernels in a QI-algebra and characterized congruence kernels whenever a QI-algebra is right distributive.

###### Strong ideals and horizontal ideals in pseudo-BCH-algebras

## Abstract

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.

###### Pre-semihyperadditive Categories

## Abstract

In this paper we extend the notion of classical (pre-)semiadditive category to (pre-)semihyperadditive category. Algebraic hyperstructures are algebraic systems whose objects possessing the hyperoperations or multi-valued operation. We introduce categories in which for objects *A* and *B*, the class of all morphisms from *A* to *B* denoted by *Mor*(*A, B*), admits an algebraic hyperstructures, such as semihypergroup or hypergroup. In this regards we introduce the various types of pre-semihyperadditive categories. Also, we construct some (pre-)semihyperadditive categories by introducing a class of hypermodules named general Krasner hypermodules. Finally, we investigate some properties of these categories.

###### Engel, Nilpotent and Solvable BCI-algebras

## Abstract

In this paper, we define the concepts of Engel, nilpotent and solvable *BCI*-algebras and investigate some of their properties. Specially, we prove that any *BCK*-algebra is a 2-Engel. Then we define the center of a *BCI*-algebra and prove that in a nilpotent *BCI*-algebra *X*, each minimal closed ideal of *X* is contained in the center of *X*. In addition, with some conditions, we show that every finite *BCI*-algebra is solvable. Finally, we investigate the relations among Engel, nilpotent and solvable *BCI*(*BCK*)-algebras.

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Cubic Intuitionistic Structures Applied to Ideals of *BCI*-Algebras

## Abstract

In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic *p*-ideals and cubic intuitionistic *a*-ideals in *BCI*-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic *q*-ideals, cubic intuitionistic *p*-ideals and cubic intuitionistic *a*-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic *p*-ideal are provided. Characterizations of a cubic intuitionistic *a*-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic *a*-ideal is established.

###### Relation Between Be-Algebras and G-Hilbert Algebras

## Abstract

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this paper, we investigate the relationship between g-Hilbert algebras, gi-algebras, implication gruopoid and BE-algebras. In fact, we show that every g-Hilbert algebra is a self distributive BE-algebras and conversely. We show cannot remove the condition self distributivity. Therefore we show that any self distributive commutative BE-algebras is a gi-algebra and any gi-algebra is strong and transitive if and only if it is a commutative BE-algebra. We prove that the MV -algebra is equivalent to the bounded commutative BE-algebra.

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Some properties of n-dimensional (∈_{γ}, ∈_{γ}, ∨_{qδ})-fuzzy subalgebra in BRK-algebras

## Abstract

The purpose of this paper is to initiate the concept of n-dimensional (∈_{γ}, ∈_{γ}, ∨_{qδ})-fuzzy subalgebra in BRK-algebra and investigate some of their related properties. We also show that the relationship between n-dimensional (∈_{γ}, ∈_{γ}, ∨_{qδ})-fuzzy subalgebra and the crisp subalgebra in BRK-algebra are discussed.