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In this paper, we introduce the notion of states and state operators on residuated skew lattices and investigate some related properties of them. The relationships between state operators and states on residuated skew lattices are discussed. We prove that every Bosbach state on a residuated skew lattice is a Riecan state and with an example we show that there is a Riecan state on a residuated skew lattice which is not a Bosbach state. Also, some conditions are given for a Riecan state on residuated skew lattice to be a Bosbach state. We present different types of state residuated skew lattices, like weak, strong, simple, local and state-morphism residuated skew lattices. Any weak state-morphism residuated skew lattice is a strong state residuated skew lattice and converse is true under an extra condition.

The present study aimed to propose the similarity triangle algebra and prove the completeness of the similarity triangle logic. The similarity was defined on triangle algebra. Similarity triangle algebra is a triangle algebra endowed with a binary operation S, which verifies specific additional properties. These properties and a class of all the similarity triangle algebras form a variety which were assessed as well. In addition, the similarity IVRL-filters (S-IVRL-filters) were investigated and introduced in the similarity triangle algebras, and the similarity triangle logic $$(\mathcal {STL})$$ was presented, which is a system of several-valued logic capturing the tautologies of similarity triangle algebras to prove the completeness theorem. According to the results, $$\mathcal {STL}$$ is a conservative extension of triangle logic ( $$\mathcal {TL}$$ ).

In this paper, at first we study strong Sheffer stroke NMV-algebra. For getting more results and some classification, the notions of filters and subalgebras are introduced and studied. Finally, by a congruence relation, we construct a quotient strong Sheffer stroke NMV-algebra and isomorphism theorems are proved., Ege University Scientific Research Projects Directorate [20772], We wish to thank the reviewers for excellent suggestions that have been incorporated into the paper. This study is supported by Ege University Scientific Research Projects Directorate with the project number 20772.

A complex intuitionistic fuzzy set (CIFS) has an ability to represent the problems with intuitionistic uncertainty and periodicity, simultaneously. In this paper, we present a new framework for handling complex intuitionistic fuzzy information by combining the CIFSs with competition graphs. We first introduce the concept of complex intuitionistic fuzzy competition graphs along with its two worthwhile extensions, namely, complex intuitionistic fuzzy k-competition and complex intuitionistic fuzzy p-competition graphs. Further, we present complex intuitionistic fuzzy neighborhood graphs and m-step complex intuitionistic fuzzy competition graphs with some of their remarkable results. We also describe an application of complex intuitionistic fuzzy competition graphs along with an algorithm which finds the competition among the applicants competing for the particular jobs possessing the two-dimensional information. Moreover, we provide a comparison analysis of proposed competition graphs with existing graphs to find the validity and superiority of proposed competition graphs.

We introduce and investigate central lifting property (CLP) for orthomodular lattices as a property whereby all central elements can be lifted modulo every p-ideal. It is shown that prime ideals, maximal ideals and finite p-ideals have CLP. Also Boolean algebras, simple chain finite orthomodular lattices, subalgebras of an orthomodular lattices generated by two elements and finite orthomodular lattices have CLP. The main results of the present paper include the investigation of CLP for principal p-ideals and finite direct products of orthomodular lattices.

We consider pseudofinite MV-algebras. As a main result, we show that an infinite MV-algebra is pseudofinite if and only if it is definably well founded, improving a result of a previous paper. Moreover, we show that the theory of pseudofinite MV-algebras has a partial form of elimination of quantifiers. Further, we show that the class of pseudofinite MV-chains and the class of pseudofinite MV-algebras are not finitely axiomatizable, we give some collapsing results for pseudofinite MV-algebras, we consider relative subalgebras of pseudofinite MV-algebras, and we study ideals of pseudofinite MV-algebras.

Knowing the applications of logical algebras in various fields, such as artificial intelligence or coding theory, in this paper, we study some properties of a special class of such algebras, namely finite Wajsberg algebras. For this purpose, we give a representation theorem for finite Wajsberg algebras and give a formula for the number of non-isomorphic Wajsberg algebras of order n; also we give the total number of finite Wajsberg algebras of order n. Since a big value of n involves a lot of computations, as examples, we present and describe all finite Wajsberg algebras of order $$n\le 9$$ .

As a generalization to hesitant fuzzy linguistic term set (HFLTS), double hierarchy hesitant fuzzy linguistic term set (DHHFLTS) is presented which circumvents the weakness of HFLTS in representing complex linguistic terms. DHHFLTS has two linguistic hierarchies with the second hierarchy supplementing the primary which enables decision-makers (DMs) to represent complex linguistic terms better. Motivated by the power of DHHFLTS, in this paper, a new decision framework is presented under DHHFLTS context. Initially, a new aggregation operator called double hierarchy hesitant fuzzy hybrid aggregation (DHHFHA) operator is proposed for sensible aggregation of DMs’ preference information. Further, weights of attributes are calculated by extending statistical variance (SV) method under DHHFLTS context. Objects are prioritized by extending the popular WASPAS (weighted aggregated sum product assessment) method to DHHFLTS context. The applicability and usefulness of the proposed framework are realized by demonstrating a risk management technique (RMT) selection problem for a construction project. Finally, the superiority and weakness of the proposed framework are discussed by comparison with other methods.

In this paper, we define residuated skew lattice as non-commutative generalization of residuated lattice and investigate its properties. We show that Green’s relation 𝔻 is a congruence relation on residuated skew lattice and its quotient algebra is a residuated lattice. Deductive system and skew deductive system in residuated skew lattices are defined and relationships between them are given and proved. We define branchwise residuated skew lattice and show that a conormal distributive residuated skew lattice is equivalent with a branchwise residuated skew lattice under a condition.

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.

In this paper, we have introduced the notion of local and semilocal triangle algebras and propose the theorems that characterize these algebraic structures. Additionally, we have established the new properties of these algebraic structures and discussed the relations between local triangle algebras and some interval valued residuated lattice (IVRL)-filters, such as n-fold IVRL-extended integral filters and IVRL-extended maximal filters. The obtained results proved that the MTL-triangle algebra is a subdirect product of local triangle algebras. Moreover, a correlation was observed between the set of the dense elements and local triangle algebras. Finally, semilocal triangle algebras were introduced and assessed in detail, and an association was observed between the semilocal triangle algebras and quotient triangle algebras. submittedVersion

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.

In this paper, congruences and (order, prime and compatible) filters are introduced in Sheffer stroke basic algebras. They are interrelated with each other. Also some relationships between filters and congruences by focusing on their properties in these structures are given. In addition to these, a basic algebra is constructed by the help of compatible filter and 𝒜/ Θ (F). Moreover, a representation theorem for Sheffer stroke basic algebra by using prime filters and 𝒜/ Θ (F) is stated and proved.

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 define the right Boolean lifting property (left Boolean lifting property) RBLP (LBLP) for pseudo BL-algebra to be the property that all Boolean elements can be lifted modulo every right filter (left filter) and next we study the behavior of RBLP (LBLP) with respect to direct products of pseudo BL-algebra. We introduce some conditions, which turn out to be a strengthening and a weakling of RBLP (LBLP) respectively and which open new ways of approaching the study of the RBLP (LBLP) in pseudo BL-algebras. Keywords: Lifting property, Boolean center, (prime, maximal) filter, (maximal, local, hyper Archimedean, quasi-local, semi local) pseudo BL-algebra

In the paper, we introduce some stabilizers and investigate related properties of them in MTL-algebras.Then, we also characterize some special classes of MTL-algebras, for example, IMTL-algebras, integral MTL-algebras, Godel algebras and MV-algebras, in terms of these stabilizers. Moreover, we discuss the relation between stabilizers and several special filters (ideals) in MTL-algebras. Finally, we discuss the relation between these stabilizers and prove that the right implicative stabilizer and right multiplicative stabilizer are order isomorphic. This results also give answers to some open problems, which were proposed by Motamed and Torkzadeh in [Soft Comput, {\bf 21} (2017) 686-693].

Supply chain management (SCM) is an attractive area for research which has seen tremendous growth in the past decades. From the literature we observe that, supplier outsourcing (SO) is a highly explored research field in SCM which lacks significant scientific contribution. The major concern in SO is the decision makers’ (DMs) viewpoint which are often vague and imprecise. To better handle such imprecision, in this paper, we propose a new two-stage decision-making framework called TSDMF, which uses hesitant fuzzy information as input. In the first stage, the DMs’ preferences are aggregated using a newly proposed simple hesitant fuzzy-weighted geometry operator, which uses hesitant fuzzy weights for better understanding the importance of each DM. Following this, in the second stage, criteria weights are estimated using newly proposed hesitant fuzzy statistical variance method and finally, a new ranking method called three-way hesitant fuzzy VIKOR (TWHFV) is proposed by extending the VIKOR ranking method to hesitant fuzzy environment. This ranking method uses three categories viz., cost, benefit and neutral along with Euclid distance for its formulation. The practicality of the proposed TSDMF is verified by demonstrating a supplier outsourcing example in an automobile factory. The robustness of TWHFV is realized by using sensitivity analysis and other strengths of TSDMF are discussed by comparison with another framework.

This paper considers wireless sensor (hyper) networks by single-valued neutrosophic (hyper)graphs. It tries to extend the notion of single-valued neutrosophic graphs to single-valued neutrosophic hypergraphs and it is derived single-valued neutrosophic graphs from single-valued neutrosophic hypergraphs via positive equivalence relation. We use single-valued neutrosophic hypergraphs and positive equivalence relation to create the sensor clusters and access to cluster heads. Finally, the concept of (extended) derivable single-valued neutrosophic graph is considered as the energy clustering of wireless sensor networks and is applied this concept as a tool in wireless sensor (hyper) networks.

This paper aimed to study distributivity and Jordan-Dedekind condition in hyperlattices. Hyperlattice L satisfies the Jordan-Dedekind condition if all maximal chains between the same end points hav...

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.

This paper presents a new two-tier decision making framework with linguistic preferences for scientific decision making. The major reason for adopting linguistic preference is to ease the process of rating of alternatives by allowing decision makers (DMs) to strongly emphasize their opinion on each alternative. In the first tier, aggregation is done using a newly proposed operator called linguistic based aggregation (LBA), which aggregates linguistic terms directly without making any conversion. The main motivation for this proposal is driven by the previous studies on aggregation theory which reveals that conversion leads to loss of information and formation of virtual sets which are no longer sensible and rational for decision making process. Secondly, in the next tier, a new ranking method called IFSP (intuitionistic fuzzy set based PROMETHEE) is proposed which is an extension to PROMETHEE (preference ranking organization method for enrichment evaluation) under intuitionistic fuzzy set (IFS) context. Unlike previous ranking methods, this ranking method follows a new formulation by considering personal choice of the DMs over each alternative. The main motivation for such formulation is derived from the notion of not just obtaining a suitable alternative but also coherently satisfying the DMs’ viewpoint during decision process. Finally, the practicality of the framework is tested by using supplier selection (SS) problem for an automobile factory. The strength and weakness of the proposed LBA-IFSP framework are verified by comparing with other methods under the realm of theoretical and numerical analysis. The results from the analysis infer that proposed LBA-IFSP framework is rationally coherent to DMs’ viewpoint, moderately consistent with other methods and highly stable and robust against rank reversal issue.

In general, when block codes are defined, the used alphabet is not binary. It is used an alphabet with n elements, $$n\ge 2,$$ identified usually with the set $$ A_{n}=\{0,1,2,\ldots ,n-1\}$$ . Recently, some papers were devoted to the study of the connections between binary block codes and BCK-algebras. These codes are called n-ary block codes. In this paper, we try to generalize these results to n-ary block codes, providing an algorithm which allows us to construct a BCK-algebra from a given n-ary block code. For this purpose, we will prove that to each n-ary block code V we can associate a BCK-algebra X, such that the n-ary block code generated by $$X,V_{X},$$ contains the code V as a subset and we will find in what circumstances the converse is also true.

In this paper, we show that the going up and lying over theorems hold in PMV-algebras, and we prove that every $$\cdot $$ -prime ideal in a PMV-subalgebra is the intersection of a $$\cdot $$ -prime ideal in the overalgebra with the subalgebra. Also, we show that if P is a $$\cdot $$ -prime ideal of a unital PMV-algebra A and $$A^{\prime }$$ is a subalgebra of A, having P as a maximal $$\cdot $$ -ideal, then $$A^{\prime }/0_{P}(A)$$ is a local PMV-algebra which is called the localization of A at P relative to $$A^{\prime }$$ where $$0_{P}(A)$$ is the intersection of all $$\cdot $$ -prime ideals of A contained in P.

In this paper, we study local Hilbert algebras and give some theorems that characterize local Hilbert algebras. Also, we prove that, (i) H is a local Hilbert algebra if and only if the set of all dense elements of H is H ∖ {0}. (ii) H is a local Hilbert algebra if and only if the set of all regular elements of H is a local Boolean algebra. In a local Hilbert algebra H with supremum, we prove that: (i) If F is a Boolean filter of H, then F is a prime filter. (ii) F is a Boolean filter iff F is a maximal filter iff F = H ∖ {0}.

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.

In recent years, there has been a marked increase in the use of wireless sensor networks in various environments such as crisis areas, military operations, and monitoring systems. These networks do not use a fixed network infrastructure and therefore they are a popular choice for highly dynamic environments. One of the main concerns in these networks is the topology management issue, which the clustering method is a subfield for that. The main objective of clustering methods is optimizing the energy consumption. This paper proposes a new clustering protocol, which uses many parameters such as the activity history of each node, local and general state of nodes and their resources condition to determine the best cluster heads and members of each cluster that can increase the network lifetime, fair resource consumption and network coverage.

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined, i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work.

Nowadays, the most important aspects of wireless sensor networks (WSNs) are to make optimal use of and direct the limited energy of sensor nodes towards the desired application and prolong the network lifetime for that application. Although a few studies exist that have addressed these special goals, they have been mostly focused on the process of selecting cluster heads (CHs) and forwarders. No studies have been conducted so far on the selection of fuzzy input parameters in clustering and routing processes as well as the application-based parameter selection process. Generally, a fixed number of parameters have always been selected by designers. Hence, the shuffled frog leaping algorithm (SFLA) was employed in this paper to propose a technique for selecting fuzzy input parameters in a fuzzy multi-hop clustering protocol named the PS-SFLA. This technique includes three main phases, introduced in three versions for the sake of stepwise evaluation. Based on the literature review, the most frequent and diverse parameters were extracted and formulated in the first version. The proposed technique used the SFLA in the second version to select the appropriate parameters fitting the application specifics and scenario and determine the coefficients of parameters simultaneously so that they could be used as the inputs of the fuzzy inference system. It was also utilized in the final version for the automated, accurate, application-based tuning of fuzzy rules before the network was set up. By design, different versions of the PS-SFLA act as the starting points of the next version in addition to the fact that they can be evaluated separately. The PS-SFLA was compared with the LEACH, ASLPR, SIF, ERA, and FSFLA in different scenarios and two applications from the perspective of the alive nodes, the number of packets received by the BS, lifetime, and other factors. According to the simulation results, the PS-SFLA outperformed all of the other methods greatly in all scenarios and applications and PS-SFLA increases the lifetime due to the appropriate selection of fuzzy input parameters based on application and purpose.

In this paper, extension of relative co-annihilator in residuated lattices is introduced for subsets T and Y of residuated lattice L and is denoted by (T, Y ). In this extension, T has a more important role than Y , as expected. It is proved if T is a filter, then co-annihilator of T relative to Y is a filter and is the same as the co-annihilator of T relative to the filter generated by Y. Also some theorems are stated and proved which determine the relationship between this special subset and the other types of filters. Finally, we give some conditions for subsets T and Y of residuated lattices L, for which (T, Y ) is a special subset of L.

This paper introduces the notion of (locally) fuzzy homomorphism in fuzzy hyper BE-subalgebra and makes a connection between fuzzy BE-algebras and hyper BE-algebras via this notion. Considering the fuzzy homomorphism in fuzzy hyper BE-subalgebra, this study defines some equivalence relations in a way that, under some conditions, are regular or strongly regular and so derives fuzzy BE-algebras by quotient of hyper fuzzy BE-subalgebras. In fact, it tries to introduce some strongly regular relations on any hyper BE-algebras by using the concept of fuzzy hyper BE-subalgebras and fuzzy homomorphism. Moreover, the concept of normal fuzzy homomorphism is defined and some new normal fuzzy homomorphisms are introduced to make better connections between fuzzy BE-algebras and fuzzy hyper BE-subalgebras.

Nowadays, companies are paying more attention to supply chain management in order to achieve success in the competitions and keep the customers satisfied. Therefore, supplier selection is of prime importance in a supply chain which has a key role in determining the success or failure of a business. In this paper, fuzzy multi-objective linear programming model with group decision making is presented. Firstly, linguistic variables in the form of interval-valued intuitionistic fuzzy numbers and group decisions are defined. By using TOPSIS method, the weight of each criterion (objective) and weight of each constraint are determined. Then, with defining membership functions of the objectives and membership functions of suppliers’ constraints and using the achieved weights from the previous phase, the supplier selection problem will be converted to fuzzy multi-objective linear programming model which determines the appropriate ordering quantities for each supplier. Finally, two numerical examples for supplier selection are used to demonstrate the practicality and effectiveness of the proposed approach.

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 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 constructs code-based hypergraphs from BCH codes via positive equivalence relation and proves that all hypergraphs are BCH code-based hypergraphs. It shows that every hypergraph is isomorphic to a BCH-hypergraph. We characterize minimum distance and maximum distance of codewords in hyperedges of BCH code-based hypergraphs.

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∗).

Since Wireless Sensor Networks (WSNs) are composed of a set of sensor nodes that limit resource constraints such as energy constraints, energy consumption in WSNs is one of the challenges of these networks. One of the solutions to reduce energy consumption in WSNs is to use clustering. In clustering, cluster members send their data to their Cluster Head (CH), and the CH after collecting the data, sends them to the Base Station (BS). In clustering, choosing CHs is very important; so many methods have proposed to choose the CH. In this study, a hesitant fuzzy method with three input parameters namely, remaining energy, distance to the BS, distance to the center of cluster is proposed for efficient cluster head selection in WSNs. We define different scenarios and simulate them, then investigate the results of simulation.

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 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.