Implicative Ideals In BCK-algebras Research Articles

Implicative ideals of BCK-algebras based on Dokdo structure

A Dokdo structure is used to study implicative ideals in BCK-algebras. The notion of Dokdo implicative ideals in BCK-algebras is introduced, and the relevant properties are investigated. The relationship between Dokdo subalgebras, Dokdo ideals, and Dokdo implicative ideals are discussed, and conditions that allow a Dokdo ideal and a Dokdo subalgebra to be a Dokdo implicative ideal are provided.

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.

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.

Bipolar Fuzzy Implicative Ideals of BCK-Algebras

The notion of bipolar fuzzy implicative ideals of a BCK-algebra is introduced, and several properties are investigated. The relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal is studied. Characterizations of a bipolar fuzzy implicative ideal are given. Conditions for a bipolar fuzzy set to be a bipolar fuzzy implicative ideal are provided. Extension property for a bipolar fuzzy implicative ideal is stated.

Cubic Intuitionistic Implicative Ideals of BCK-Algebras

The cubic intuitionistic implicative ideals in BCK-algebras and the relationship between a cubic intuitionistic subalgebra, a cubic intuitionistic ideal and a cubic intuitionistic implicative ideal are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic implicative ideal are provided. Characterizations of a cubic intuitionistic implicative ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic implicative ideal is established.

New types of bipolar fuzzy ideals of BCK-algebras

The notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are introduced, and related properties are investigated. Characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are given, and several properties are discussed. Finally, we prove that if T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T.

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.

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.

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.

Implicative Ideals of BCK‐Algebras Based on the Fuzzy Sets and the Theory of Falling Shadows

Based on the theory of falling shadows and fuzzy sets, the notion of a falling fuzzy implicative ideal of a BCK‐algebra is introduced. Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals are given. Relations between fuzzy implicative ideals and falling fuzzy implicative ideals are provided.

INTUITIONISTIC FUZZINESS OF IMPLICATIVE IDEALS IN BCK-ALGEBRAS

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Aranassov is one among them. In this paper, we apply the concept of an intuitionistic fuzzy set to implicative ideals in BCK-algebras. The notion of an intuitionistic fuzzy implicative ideal of a BCK-algebra is introduced, and some related properties are investigated. An extension property for intuitionistic fuzzy implicative ideals is established. Characterizations of an intuitionistic fuzzy implicative ideal are given. Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy implicative ideal are given. Using a collection of implicative ideals, intuitionistic fuzzy implicative ideals are established.

On some computational algorithms forn-fold ideals in BCK-algebras

This paper is devoted to thefoldness theory of implicative ideals in BCK-algebras. We use the fuzzy point concept to give some characterizations of fuzzyn-fold implicative ideals and establish some relationships with variousn-fold ideals. We also construct some algorithms for studying the foldness of implicative ideals in BCK-algebras.

Fuzzy implicative ideals in BCK-algebras

In this paper, we introduce the concept of fuzzy implicative ideals in BCK-algebras and study some of their properties.

FUZZY IMPLICATIVE IDEALS IN BCK-ALGEBRAS


 
 
 An implicative ideal of a BCK-algebra is defined by lseki [1]. In 1991, Xi [7] defined a fuzzy implicative ideal of a BCK-algebra. In this paper, we investigate the properties of fuzzy implicative ideals in BCK-algebras.