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We introduce Engel elements in a BCI-algebra by using left and right normed commutators, and some properties of these elements are studied. The notion of $n$-Engel BCI-algebra as a natural generalization of commutative BCI-algebras is introduced, and we discuss Engel BCI-algebra, which is defined by left and right normed commutators. In particular, we prove that any nilpotent BCI-algebra of type $2$ is an Engel BCI-algebra, but solvable BCI-algebras are not Engel, generally. Also, it is proved that $1$-Engel BCI-algebras are exactly the commutative BCI-algebras.

In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal., Ege University Scientific Research Projects DirectorateEge University [20772], The authors would like to express their sincere thanks to the referee for their valuable suggestions and comments. This study is partially funded by Ege University Scientific Research Projects Directorate with the Project Number 20772.

In this paper, we introduce the notion of a wRM/eRM-algebra as a generalization of a RM-algebra. These structures are studied in details. Also, we derived an eRM-algebra from a RM-algebra and vice versa. The concept of a positive implicative eRM-algebra is defined and we discussed on the medial filters.

In this paper, we introduce Sheffer Stroke UP-algebra (in short, SUP-algebra) and study its properties. We demonstrate that the Cartesian product of two SUP-algebras is a SUP-algebra. After presenting SUP-subalgebras, we define SUP-homomorphisms between SUP-algebras. © 2021 Tahsin Oner et al., published by Sciendo.

The aim of this study is to introduce fuzzy filters of Sheffer stroke Hilbert algebra. After defining fuzzy filters of Sheffer stroke Hilbert algebra, it is shown that a quotient structure of this algebra is described by its fuzzy filter. In addition to this, the level filter of a Sheffer stroke Hilbert algebra is determined by its fuzzy filter. Some fuzzy filters of a Sheffer stroke Hilbert algebra are defined by a homomorphism. Normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra and the relation between them are presented. By giving the Cartesian product of fuzzy filters of a Sheffer stroke Hilbert algebra, various properties are examined. © 2021 - IOS Press. All rights reserved., Ege Üniversitesi: 20772, The authors are thankful to the referees for a careful reading of the paper and for valuable comments and suggestions. This study is partially funded by Ege University Scientific Research Projects Directorate with the Project Number 20772.

A complex Pythagorean fuzzy set, an extension of Pythagorean fuzzy set, is a powerful tool to handle two dimension phenomenon. Dombi operators with operational parameters have outstanding flexibility. This article presents certain aggregation operators under complex Pythagorean fuzzy environment, including complex Pythagorean Dombi fuzzy weighted arithmetic averaging (CPDFWAA) operator, complex Pythagorean Dombi fuzzy weighted geometric averaging (CPDFWGA) operator, complex Pythagorean Dombi fuzzy ordered weighted arithmetic averaging (CPDFOWAA) operator and complex Pythagorean Dombi fuzzy ordered weighted geometric averaging (CPDFOWGA) operator. Moreover, this paper explores some fundamental properties of these operators with appropriate elaboration. A decision‐making numerical example related to the selection of bank to purchase loan is given to demonstrate the significance of our proposed approach. Finally, a comparative analysis with existing operators is given to demonstrate the peculiarity of our proposed operators.

The present study aimed to investigate the similarity monadic $BL$-algebras and prove the completeness of the similarity monadic basic logic. To this end, the similarity monadic $BL$-algebras were defined as: a variety of monadic $BL$-algebras equipped with unary operators $\forall$ and $\exists$, and the vital properties of the algebras were provided. In addition, the S-monadic filters in the similarity monadic $BL$-algebras were introduced and investigated. Correspondingly, the representable similarity monadic $BL$-algebras were characterized, and the notion of the logic of the similarity monadic $BL$-algebras were introduced to prove their soundness and completeness.

In this paper, we introduce the notion of implication filter as a generalization of Boolean filter of first kind in Hilbert algebra and study it in detail. Also, we prove that F is an implication filter of a Hilbert algebra H if and only if every implicative filter of quotient algebra H / F is an implication filter. Finally, we generalized Boolean algebra and introduced a weak implication algebra, we prove that F is an implication filter of a Hilbert algebra H if and only if the quotient algebra H / F is a weak implication algebra. By suitable diagrams, we summarize the results of this paper and the previous results in these fields.

This paper introduces and applies the notion of fuzzy hyper EQ-subalgebras as a generalization of the notation of fuzzy EQ-subalgebras and constructs extendable fuzzy EQ-subalgebras via these concepts. Moreover, by using the fuzzy hyper EQ-filters, it defines a congruence relation on hyper EQ-algebras that under some conditions is strongly regular and the quotient of any hyper EQ-algebra via this relation is an EQ-algebra.

In this paper, we introduce the notion of pseudo-CI algebras and investigate some of their properties. It is a generalization of the notion of pseudo-BE algebras, pseudo-BCK algebras and pseudo-MV algebras. We give and provide some conditions for a pseudo-CI algebra to be a pseudo-BE algebra. Also, we define the class of singular pseudo-CI algebras and show that any singular pseudo-CI algebra is a pseudo-BCI algebra. The adjoint group of a pseudo-CI algebra is defined and studied. Furthermore, distributive pseudo-CI algebras are discussed and showed that these coincide with distributive pseudo-BE algebras.

In this paper, we introduce the notions of stabilizer of a subset and the stabilizer of a subset with respect to another one in triangle algebras and study them in details. It is shown that the stabilizer of a subset and stabilizer of an interval valued residuated lattice filter (IVRL-filter) with respect to another IVRL-filter are IVRL-filters. We state and prove some theorems which determine some properties of this stabilizers in triangle algebras. Also, we prove that in linearly ordered triangle algebras, stabilizer of a set is an IVRL-extended prime filter. Finally, we consider the influence of stabilizers on product and quotient triangle algebras.

The notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

In this paper, we introduce the notions of multiplier, ⊙ -closure operator and modal operator on hoop-algebras. After that, we investigate some algebraic properties of multiplier, ⊙ -closure operator and modal operator on hoop-algebras. Then we study relationships between multiplier and closure operator on hoop-algebras showing that, any monotone modal operator on hoop-algebra serves as a closure operator. Finally we represent image of hoop-algebra under the closure operator can be a hoop-algebra.

In this paper, the notions of commutators and pseudo commutators of elements (subsets) of a BCI-algebra are introduced and some properties are given. The concept of solvable BCI-algebras are also discussed and their properties are investigated. It is proved that the class of solvable BCI-algebras is closed under sub-algebra, cartesian product and inverse image operations.

The purpose of this paper is the study of prime ideals and prime filters in hyperlattices. I-filter and the filter generated by a ∈ L are introduced. Moreover, we introduce dual distributive hyperlattices, and I-filter in dual distributive hyperlattices. Some properties of hyperlat- tices are studied and the relationship between prime ideals and prime filters in hyperlattices is discussed.

The aim of this work is to introduce some types of filters in Hilbert algebras. Some theorems are stated and proved which determine the relationship between these notions and other filters of Hilbert algebra and by some examples we show that these concepts are different. The relationships between these filters and quotient algebras that are constructed via these filters are described.

The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized and an analogous of representation theorem for very true MTL-algebras is proved. Then, the left and right stabilizers of very true MTL-algebras are introduced and some related properties are given. As applications of stabilizer of very true MTL-algebras, we produce a basis for a topology on very true MTL-algebras and show that the generated topology by this basis is Baire, connected, locally connected and separable. Finally, the corresponding logic very true MTL-logic is constructed and the soundness and completeness of this logic are proved based on very true MTL-algebras.

This paper introduces the notion of fuzzy hyper EQ-subalgebras as generalization of the notation of fuzzy EQ-subalgebras and constructs extendable fuzzy EQ-subalgebras via these concepts. Moreover, we investigated some properties in fuzzy EQ-subalgebras, fuzzy hyper EQ-subalgebras by their α-level sets and connect them via positive fundamental relation C(σ∗).

This article will introduce K-modal BL-logic and investigate some properties of this new logic. Consequently, T -modal BL-logic and S4-modal BL-logic, as schematic extensions of K-modal BL-logic, will be dealt with. Likewise, the algebraic semantics based on each logic will be constructed. Finally, the soundness and completeness theorems of these logics will be proved.

In this paper, we are going to introduce a fundamental relation "δ" on (hyper BE-algebra) dual hyper K-algebra and investigate some properties. We show that quotient of any dual hyper K-algebra via a regular relation is a hyper BE-algebra and this quotient, via any strongly regular relation is a BE-algebra. Furthermore, it shows that "δ" under some condition is transitive and quotient of any weak commutative dual hyper K-algebra via "δ ∗ " is a dual BCK-algebra.

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.

The concept of neutrosophic set (NS) developed by Smarandache is a more general platform which extends the concepts of the classic set and fuzzy set , intuitionistic fuzzy set (see [1]) and interval valued intuitionistic fuzzy set .

In this paper, we first introduce the notion of implication filters in BL-algebras and then state and prove some theorems which determine the relationships of these filters and other filters in BL-algebras. We also introduce the notion of implication BL-algebras and investigate its properties, as well as the relationship of implication BL-algebras and other algebraic structures.

In this paper, we introduce the notion of upper (down) set in BL-algebras and extended upper sets of BL-algebras. Also, the notions of Krull dimension and regular sequence in BL-algebras are introduced and studied in detail.

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.

In this paper, we introduce the concept of TL-filter of BL-algebra and study some basic properties. TL-filter generated by L-subsets and the concept of T-congruence L-relations on BL-algebra are introduced and the relationships between TL-filters and T-congruence L-relations on BL-algebra are studied.

In this article, we introduced the notions fuzzy soft subalgebras and fuzzy soft ideal of subtraction algebras. We showed that fuzzy soft image and fuzzy soft inverse image of fuzzy soft ideal are both fuzzy soft ideal under certain conditions. We have studied their related properties in details.

In this paper, generalization of the fuzzy BCK/BCI-algebras of type (α, β)* are introduced. Then we state and prove some theorems that determine the relationship between these notions and subalgebras of BCK/BCI-algebras. Characterizations of (α, β)*-fuzzy subalgebras in a BCK/BCI-algebras are given.

In this paper, we define the Smarandache hyper BCC-algebra, and Smarandache hyper BCC-ideals of type 1, 2, 3 and 4. We state and prove some theorems in Smarandache hyper BCC-algebras, and then we determine the relationships between these hyper ideals.

In this paper the notion of fuzzy BE-algebras are introduced. We state and prove some theorems in fuzzy BE-algebras and level subalgebras. Some characterizations of fuzzy subalgebras of BE-algebras are given. Using fuzzy subalgebras, characterizations of Artinian and Noetherian BE-algebras are also established. Finally the notion of fuzzy topological BE-algebras is introduced. We state and prove the Foster’s results on homomorphic images and inverse images in fuzzy topological BE-algebras.

In this paper we define the Smarandache BL -algebra, Q -Smarandache ideal and Q -Smarandache implicative ideal, we obtain some related results. After that by considering the notions of these ideals we determine relationships between ideals in BL -algebra and Q -Smarandache (implicative) ideals of BL -algebra. Finally we construct quotient of Smarandache BL -algebras via MV -algebra and prove some theorems.

In this paper the notion of fuzzy BM-algebra and fuzzy topological BM-algebras are introduced. We stated and proved some theorem in fuzzy BM-algebras and level subalgebras. Finally fuzzy topological BM-algebras are studied.

In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse.

In this note we first define the notions of (weak, strong) implicative hyper K-algebras. Then we show by examples that these notions are different. After that we state and prove some theorems which determine the relationship between these notions and (weak) hyper K-ideals. Also we obtain some relations between these notions and (weak) implicative hyper K-ideals. Finally, we study the implicative hyper K-algebras of order 3, in particular we obtain a relationship between the positive implicative hyper K-algebras and (weak, strong) implicative hyper K-algebras under a simple condition.

In this paper we introduce (normal, implicative, maximal, involutive) ideals of BZMV dm -algebras. Also the ideal generated by a set in BZMV dm-algebra are introduced and the relationships between them are studied.

In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE- algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse. Finally, we state and prove the relationship between hesitant fuzzy (implicative) filters and γ-inclusive sets.

In this paper we introduced notion of fuzzy filters and fuzzy ideals on CI–algebras. We give the construction of quotient (Xθ;o,1θ) via a fuzzy congruence relation of CI–algebra X.