Implicative Ideals Research Articles

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.

Smarandache BL-algebra

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.

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.

Ideals of BCI-algebras and their Properties

Ideals of BCI-algebras and their PropertiesIn this article three classes of ideals are discussed: associative ideals, commutative ideals, implicative ideals and positive implicative ideals, and their elementary properties. Some of their properties and the relationships between them have not been proven yet, and will be completed in the following article.MML identifier: BCIIDEAL, version: 7.8.10 4.99.1005

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 SET THEORY APPLIED TO IMPLICATIVE IDEALS IN BCK-ALGEBRAS

As a continuation of [4], characterizations of fuzzy implicative ideals are given. An extension property for fuzzy implicative ideals is established. We prove that the family of fuzzy implicative ideals is a completely distributive lattice. Using level subsets of a BCk-algebra X with respect to a fuzzy set <TEX>$\={A}$</TEX> in X, we construct a fuzzy implicative ideal of X containing <TEX>$\={A}$</TEX>.

Fuzzy ideals in BCI-algebras

The aim of this paper is to introduce the notions of fuzzy BCI-positive implicative ideals and fuzzy BCI-implicative ideals of BCI-algebras and to investigate their properties. We give several characterizations of fuzzy BCI-positive implicative ideals and fuzzy BCI-implicative ideals. The extension theorems of fuzzy BCI-positive implicative ideals and fuzzy BCI-implicative ideals are obtained. We discuss the relations among various fuzzy ideals and show that a fuzzy set μ of a BCI-algebra is fuzzy BCI-implicative ideal if and only if μ is both fuzzy BCI-positive implicative ideal and fuzzy BCI-commutative ideal. Finally, we characterize implicative (positive implicative) BCI-algebras via fuzzy BCI-implicative (fuzzy BCI-positive implicative) ideals.

On n‐fold fuzzy positive implicative ideals of BCK‐algebras

We consider the fuzzification of the notion of an n‐fold positive implicative ideal. We give characterizations of an n‐fold fuzzy positive implicative ideal. We establish the extension property for n‐fold fuzzy positive implicative ideals, and state a characterization of PIn‐Noetherian BCK‐algebras. Finally we study the normalization of n‐fold fuzzy positive implicative ideals.

On (*)-ideals and positive implicative ideals in bci-algebras

In the present paper, we given sorne equivalent conditions of (*)-ideals and positive implicative ideals in BCI-algebras and we clarify the relation of the two classes of ideals. Also, we obtain further properties of these ideals. Finally, we discuss a quotient algebra of a BCI-algebra by a closed (*)-ideal and a quotient algebra by positive implicative ideal.

Fuzzy implicative ideals of BCK-algebras

The notion of fuzzy implicative ideals of BCK-algebras is introduced. The following results are obtained: for a BCK-algebra X, and fuzzy ideal μ of X must be a fuzzy subalgebra of X; suppose that μ and v are fuzzy ideals of X with μ ⪕ v and μ(0) = v(0); if μ is a fuzzy implicative ideal of X the so is v; a fuzzy ideal μ of X is fuzzy implicative if and only if μ is both fuzzy positive implicative and fuzzy commutative; a BCK-algebra X is implicative (resp. positive implicative, commutative) if and only if every fuzzy ideal of X is fuzzy implicative (resp. fuzzy positive implicative, fuzzy commulative). Also, several characterizations of fuzzy implicative ideals are given.

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

&#x0D; &#x0D; &#x0D; 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. &#x0D; &#x0D; &#x0D;

Fuzzy implicative and Boolean ideals of MV-algebras

Abstract We show how to obtain results in MV -algebras by considering their fuzzy ideals. In particular we prove the following theorem: Let μ be a fuzzy ideal of an MV -algebra X . Then the folllowing are equivalent: (1) μ (x ∧ x = μ(0) for all x ϵ X , (2) μ ( x n ) = μ ( x ) for all x ϵ X and all n ⩾ 1, (3) X / X μ is a Boolean algebra, (4) μ is fuzzy implicative. If I is an ideal of X and we take μ = χ I then we obtain corresponding results for the ideal I and X . We show how to extend fuzzy prime and fuzzy implicative ideals. Various fuzzy radicals are considered and related to the corresponding radicals of the algebra, and we show how they live in the spectrum of the algebra.