Positive Implicative Ideals Research Articles

Bipolar Fuzzy Commutative Ideals in BCK-algebras

This paper presents the concept of the notions of bipolar fuzzy commutative ideals of a BCK-algebra. We also study various properties regarding this concept. We prove various characterisations for bipolar fuzzy commutative ideals. Moreover, a relationship between bipolar fuzzy commutative ideals and bipolar fuzzy positive implicative ideals is presented. In addition, we present the notation of bipolar fuzzy characteristic commutative ideal. Then a relationship between a bipolar fuzzy characteristic commutative ideal and its level-cut commutative ideals is provided.

Multi-polar Q-hesitant Fuzzy Soft Implicative and Positive Implicative Ideals in BCK/BCI-algebras.

This paper focuses on exploring restricted mathematical concepts within the domain of BCK/BCI-algebras, specifically delving into the intricate realm of Multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. BCK and BCI-algebras are pivotal structures in mathematical logic and algebraic systems, finding widespread applications in fields like computer science and artificial intelligence. Our contribution lies in the introduction and thorough investigation of the innovative notions of multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals, uniquely tailored for BCK/BCI-algebras. These ideals exhibit exceptional flexibility in managing uncertain and hesitant information, serving as potent tools for modeling and solvingreal-world problems characterized by imprecise or incomplete data. This study rigorously defines and explores the foundational properties of multi-polar Q-hesitant fuzzy soft implicative ideals, underscoring their relevance and applicability within BCK/BCI-algebras. Additionally, we present the concept of positive implicative ideals, establishing their interconnectedness with multi-polar Q-hesitant fuzzy soft implicative ideals. Our investigation delves into these ideals’ algebraic and logical facets, offering valuable insights into their mutual interactions and significance within the context of BCK/BCI-algebras. To facilitate practical implementation, we develop algorithms and methodologies for identifying and characterizing multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. These computational tools enable efficient decision-making in scenariosinvolving uncertainty. Through illustrative examples and case studies, we showcase the potential of these ideals in handling complex, uncertain information, demonstrating their efficacy in aiding problem-solving processes. This research contributes significantly to advancing BCK/BCI-algebra theory by introducing innovative mathematical structures that bridge the gap between fuzzy logic, soft computing, and implicative ideals. The proposed multi-polar Q-hesitant fuzzy soft implicativeand positive implicative ideals open new avenues for addressing real-world problems characterized by imprecision and uncertainty. As such, they represent a valuable addition to the field of algebraic structures and their applications.

Cubic Of Positive Implicative Ideals In KU- Semigroup

In this paper, we define a cubic positive implicative-ideal, a cubic implicative-ideal and a cubic commutative-ideal of a semigroup in KU-algebra as a generalization of a fuzzy (positive implicative-ideal, an implicative-ideal and a commutative-ideal) of a semigroup in KU-algebra. Some relations between these types of cubic ideals are discussed. Also, some important properties of these ideals are studied. Finally, some important theories are discussed. It is proved that every cubic commutative-ideal, cubic positive implicative-ideal, and cubic implicative-ideal are a cubic ideal, but not conversely. Also, we show that if Θ is a cubic positive implicative-ideal and a cubic commutative-ideal then Θ is a cubic implicative-ideal. Some examples of the opposite direction of the previous theories are obtained.

Doubt m-Polar Fuzzy Sets Based on BCK-Algebras

Doubt m-polar subalgebras (ideals) were introduced and some properties were investigated. Also, doubt m-polar positive implicative (commutative) ideals were defined and related results were proved.

Lukasiewicz fuzzy positive implicative ideals in BCK-algebras

In BCK-algebras, the notion of Lukasiewicz fuzzy positive implicative ideal is introduced, and several properties are investigated. The relationship between Lukasiewicz fuzzy ideal and Lukasiewicz fuzzy positive implicative ideal is discussed, and characterizations of a Lukasiewicz fuzzy positive implicative ideal are considered. Conditions for a Lukasiewicz fuzzy ideal to be a Lukasiewicz fuzzy positive implicative ideal are provided, and conditions for the ∈-set, q-set and O-set to be positive implicative ideals are explored.

Positive implicative ideals of BCK-algebras based on neutrosophic sets and falling shadows

Positive implicative ideals of BCK-algebras based on neutrosophic sets and falling shadows

Interval-Valuedm-Polar Fuzzy Positive Implicative Ideals inBCK-Algebras

In this paper, the notion of interval-valuedm-polar fuzzy positive implicative ideals in BCK-algebras is presented. Then, the relationships between interval-valuedm-polar fuzzy positive implicative ideals and interval-valuedm-polar fuzzy ideals are investigated. After that, the concepts of interval-valuedm-polar∈,∈∨qκ˜-fuzzy positive implicative ideals and interval-valuedm-polar∈,∈ ∨qκ˜-fuzzy ideals are defined and some equivalent conditions are provided. Furthermore, we show that interval-valuedm-polar∈,∈ ∨qκ˜-fuzzy positive implicative ideals are interval-valuedm-polar∈,∈ ∨qκ˜-fuzzy ideals, but the converse need not be true in general and an example is given in this aim.

Positive Implicative Neutrosophic Quadruple BCK-Algebras and Ideals

Given a set [Formula: see text], conditions for the neutrosophic quadruple [Formula: see text]-set to be a (positive implicative) neutrosophic quadruple [Formula: see text]-algebra are considered. The notion of (positive implicative) neutrosophic quadruple ideal is introduced, and the extended neutrosophic quadruple [Formula: see text]-algebra is established. Several properties are investigated.

Generalized m-Polar Fuzzy Positive Implicative Ideals of BCK-Algebras

This study focuses on combining the theories of m -polar fuzzy sets over BCK -algebras and establishing a new framework of m -polar fuzzy BCK -algebras. In this paper, we define the idea of m -polar fuzzy positive implicative ideals in BCK -algebras and investigate some related properties. Then, we introduce the concepts of m -polar ∈ , ∈ ∨ q -fuzzy positive implicative ideals and m -polar ∈ ¯ , ∈ ¯ ∨ q ¯ -fuzzy positive implicative ideals in BCK -algebras as a generalization of m -polar fuzzy positive implicative ideals. Several properties, examples, and characterization theorems of these concepts are considered.

Positive implicative MBJ-neutrosophic ideals of $BCK/BCI$-algebras

Positive implicative MBJ-neutrosophic ideals of $BCK/BCI$-algebras

Positive Implicative Ideals of BCK-Algebras Based on Intuitionistic Falling Shadows

The concepts of a positive implicative ( ∈ , ∈)-intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are introduced, and several properties are investigated. Characterizations of a positive implicative ( ∈ , ∈)-intuitionistic fuzzy ideal are obtained, and relations between a positive implicative ( ∈ , ∈)-intuitionistic fuzzy ideal and an intuitionistic fuzzy ideal are discussed. Conditions for an intuitionistic fuzzy ideal to be a positive implicative ( ∈ , ∈)-intuitionistic fuzzy ideal are provided, and relations between a positive implicative ( ∈ , ∈)-intuitionistic fuzzy ideal, a falling intuitionistic fuzzy ideal and a positive implicative falling intuitionistic fuzzy ideal are considered. Conditions for a falling intuitionistic fuzzy ideal to be positive implicative are given.

Cubic Commutative Ideals of $BCK$-algebras

In this paper, we apply the concept of cubic set to commutative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic commutative ideals, cubic subalgebras, and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic commutative ideal. We define inverse images of cubic commutative ideals and establish how the inverse images of a cubic commutative ideal becomes a cubic commutative ideal. We introduce products of cubic $BCK$-algebras. Finally, we discuss the relationships between (cubic) commutative ideals, implicative ideals, and positive implicative ideals in $BCK/BCI$-algebras.

Cubic Fuzzy Positive Implicative Extension Ideals in BCK-Algebras

An interval-valued fuzzy set denotes a fuzzy set whose membership function is many -valued and form an interval in the member ship scale. The notion of cubic sets as an extension of fuzzy set in which not only a membership degree is given but also a non-membership degree is involved. In this present study, we introduce the concept of cubic fuzzy positive implicative ide als, cubic fuzzy positive implicative extension ideals in BCK-algebras and investigate some of its properties.

EI, involutory and EQI-ideal in bounded BCK-algebras

The aim of this work is to investigate the relationship between ideals in bounded BCK-algebras so we introduce the concepts of involutory and EI-ideals in bounded BCK-algebras and characterise their properties. Also we introduce the concepts of EQI-algebras and EQI-ideals in bounded BCK-algebras and show that EQI-algebras include some important BCK structures such as involutory BCK-algebras, commutative and PC-lattices. The relationships between these ideals and quotient algebras that are constructed via these ideals are described. We clarify that EI, involutory and commutative ideals coincide in PC-lattices, whereas they are not the same in bounded BCK-algebras in general. It is proved that EQI-ideals contain some current ideals such as involutory, commutative, positive implicative and implicative ideals.

On N-Fold Positive Implicative Artinian and Positive Implicative Notherian BCK-Algebras

The notion of BCK-algebras was initiated by Imai and Iseki in 1966 as a generalization of both classical and non-classical positional calculus. In 1999, Huang and Chen introduced the notion of n-fold positive implicative ideals in BCK-algebras. In 2011, Satyanarayana and Durga Prasad introduced foldness of intuitionistic fuzzy positive implicative ideals in BCK-algebras. In this paper, we introduce the notion of n-fold positive implicative ideals, n-fold positive implicative Artinian (shortly, n PI -Artinian) and n-fold positive implicative Noe-therian (shortly, n PI -Noetherian) BCK-algebras and study some of its properties.

Some properties of (α, β)-fuzzy positive implicative ideals in BCK-algebras

In this paper, by using the concept of belongingness (∈) and quasi-coincidence (q) between fuzzy points and fuzzy sets, we introduce (α, β)-fuzzy positive implicative ideals in BCK-algebras where α, β are any of {∈, q, ∈ V q, ∈ ˄ q} with α ≠ ∈ ˄ q.

Positive Implicative Ideals of BCK-Algebras Based on Intersectional Soft Sets

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of int-soft positive implicative ideals is introduced, and related properties are investigated. Relations between an int-soft ideal and an int-soft positive implicative ideal are established. Characterizations of an int-soft positive implicative ideal are obtained. Extension property for an int-soft positive implicative ideal is constructed. The∧-product and∨-product of int-soft positive implicative ideals are considered, and the soft intersection (resp., union) of int-soft positive implicative ideals is discussed.

CHARACTERIZATIONS OF FALLING FUZZY IDEALS

Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals and falling fuzzy commutative ideals are considered. Characterizations of falling fuzzy positive implicative ideals and other related ideals are discussed.

Fuzzy -Fold BCI-Positive Implicative Ideals in BCI-Algebras

We consider the notion of fuzzy -fold positive implicative ideals in BCI-algebras. We analyse many properties of fuzzy -fold positive implicative ideals. We also establish an extension properties for fuzzyn-fold positive implicative ideals in BCI-algberas. This work generalizes the corresponding results in the crisp case.

Fuzzy positive implicative ideals of BCK-algebras based on the theory of falling shadows

On the basis of the theory of a falling shadow, the notion of a falling fuzzy positive implicative ideal of a BCK-algebra is introduced. Relations between falling fuzzy positive implicative ideals and falling fuzzy ideals are given. Relations between fuzzy positive implicative ideals and falling fuzzy positive implicative ideals are provided.