We ‎define and study the pseudo BI-algebras as a generalization of BI-algebras and implication algebras and investigate some properties‎. Also‎, ‎we define distributive pseudo BI-algebras and construct a BI-algebra related to these‎. ‎Further‎, ‎we prove ‎there is no proper pseudo BI-algebra of the order less than 4 and that every pseudo BI-algebra of order 4 is a poset‎, ‎and so is a pseudo BH-algebra‎. ‎‎Beside‎, ‎we introduce exchangeable pseudo BI-algebra and show that the class of them is a proper subclass of the class pseudo CI-algebras‎. ‎Finally‎, ‎we define the notions of (weak) commutative pseudo BI-algebras and prove ‎every weak commutative pseudo BI-algebra is a (dual) pseudo BH-algebra‎, ‎but the converse is not true‎, ‎and show that every exchangeable commutative pseudo BI-algebra is an implication algebra.

‎The notion of a (branchwise) commutative $BI$-algebra is presented, and some related properties are investigated. We show that the class of commutative $BH$-algebras is broader than the class of commutative $BI$-algebras. Moreover, we %show that prove every singular $BI$-algebra is a $BH$-algebra. Also, we define the commutative ideals in $BI$-algebras and characterize the commutative $BI$-algebras in terms of commutative ideals.

This paper, applies the concept of KM -fuzzy metric spaces and introduces a novel concept of KM -fuzzy metric graphs based on KM -fuzzy metric spaces. This study, investigates on the finite KM -fuzzy metric spaces with respect to metrics and KM -fuzzy metrics and constructs KM -fuzzy metric spaces on any given non-empty sets. It tries to extend the concept of KM -fuzzy metric spaces to a larger class of KM -fuzzy metric spaces such as union and product of KM -fuzzy metric spaces and in this regard investigates on a class of products of KM -fuzzy metric graphs.

In this paper, the notion of a medial filter in a BE-algebra is defined, and the theory of filters in BE-algebras is developed. These filters are very important for the study of congruence relations in BE-algebras. Moreover, the relationships between implicative filters, medial filters and normal filters are investigated.

This paper, applies the concept of KM-fuzzy metric spaces and introduces a novel concept of KM-fuzzy metric graphs based on KM-fuzzy metric spaces. This study, investigates the finite KM-fuzzy metric spaces with respect to metrics and KM-fuzzy metrics and constructs KM-fuzzy metric spaces on any given non-empty sets. It tries to extend the concept of KM-fuzzy metric spaces to a larger class of KM-fuzzy metric spaces such as union and product of KM-fuzzy metric spaces and in this regard investigates a class of products of KM-fuzzy metric graphs.

This paper, applies the concept of KM-fuzzy metric spaces and introduces a novel concept of KM-fuzzy metric hypergraphs based on KM-fuzzy metric spaces. In special cases, we add some conditions to axioms of KM-fuzzy metric hypergraphs(to obtain of elementary hypergraphs, C-accessible hypergraphs, Cor-able hypergraphs, fuzzy hypergraphs) and so obtain locally strong KM-fuzzy metric hypergraphs and strong KM-fuzzy metric hypergraphs. This study, investigates on the finite KM-fuzzy metric spaces with respect to metrics, KM-fuzzy metrics and constructs KM-fuzzy metric spaces on any given non-empty sets. It tries to extend the concept of KM-fuzzy metric spaces to union of KM-fuzzy metric spaces and product of KM-fuzzy metric spaces and in this regard investigates on union and product of KM-fuzzy metric hypergraphs.

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition (S) and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that any PC-lattice with condition (S) is a distributive BCK-algebra.

In this article we introduce the notion of e-group as a new generalization of a group. The condition for a group to be an e-group is given. The characterization of some properties is established and some results follow.

Hoop-algebra is a naturally ordered commutative residuated integral monoids, which is introduced by B. Bosbach in Refs. 7 and 8. Now, in this paper, by considering the notion of hyper hoop-algebra, we find a condition to obtain a hoop from a finite bounded hyper hoop-algebra. Then we investigate the relations among hyper hoop-algebras and some of the other logical (hyper) algebras such as (hyper) [Formula: see text]-algebras, hyper [Formula: see text]-algebras, hyper [Formula: see text]-algebras, hyper [Formula: see text]-algebras and hyper [Formula: see text]-algebras.

The paper is devoted to introduce the notions of PMTL filters, Rl filters and PBL filters in residuated lattices and to investigate their properties. Several characterizations of left-(right-) MTL filters, left-(right-) Rl filters and left-(right-) BL filters are derived.

In this paper, we introduce the notion of sBCI/sBCK/eBCI/eBCK-algebras as a generalization of the notion of BCI/BCK-algebras. This structure is studied in detail. Also we introduce a way to make an eBCK-algebra from a BCK-algebra and vice versa.

Sh. Ghorbani, et al. [9], generalized the concept of MV -algebras and defined the notion of hyper MV -algebras. Now, in this paper, we try to prove that any hyper MV -algebra is a hyperlattice. First we prove that any hyper MV -algebra that satisfies the semi negation property is a hyperlattice. Then with a computer program, we show that any hyper MV -algebra of order less than 6, is a hyperlattice. Finally, we claim that this result is correct for any hyper MV -algebra.

In this paper, we introduce a new algebra, called an eBE-algebra, which is a generalization of a BE-algebra and discuss its basic properties. Also, the notion of filters in this structure is studied. We show that every filter can state as a union of extension of upper sets.

In this paper by considering of congruence relations induced by fuzzy ideals, we study rough sets in BCK-algebras. To this purpose we clarify a lower and upper approximations for any subset of X in this algebras. Finally, lower and upper rough ideals with respect to fuzzy ideal μ of X are discussed .

In this paper we introduce and characterise the concepts of pseudo topological hyper Kalgebras, strong pseudo topological hyper K-algebras and topological hyper K-algebras. Then we find some properties of this structures. Also we define a topology on a hyper K-algebra, which makes a pseudo topological hyper K-algebra.

The purpose of this paper is the study of algebraic properties of soft sets in a BCH-algebras. In this regards we introduce and study soft ideals and idealistic soft BCH-algebras.

In this paper, we characterize all \(B\)-algebras of order less than \(10\) and Cayley tables of them are represented. We prove that there is not any proper \(B\)-algebra of odd number with order less than \(10\).

In this paper, we introduce commutative hyper BE-algebra and study it in detail. We show that every commutative (row diagonal, column row, very thin) hyper BE-algebra is a BE-algebra.

Abstract In this paper, we introduce the notion of distributive pseudo BE-algebra and show that the related relation defined on this structure is transitive and prove that every pseudo upper set is a pseudo filter. Also, the pseudo filter generated by a set is define and show that the set of all pseudo filters is distributive complete lattice but it is not complemented. the notion of prime and irreducible subset and prove that every irreducible subset is prime.

In this paper, we introduce the notion of hyper BE–algebra and investigate some properties. Also, some types of hyper filters in hyper BE–algebras are studied and the relationship between them are stated. We try to show that these notions are independent by some examples. Furthermore, it shows that under special condition hyper BE–algebras are equivalent to dual hyper K–algebras. AMS Mathematics Subject Classification (2010): 06F35, 03G25.