• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.541-544
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• 1995

In [1], Ahmad has given a characterization of prime dual ideals in bounded commutative BCK-algebras. The aime of this paper is to show that Theorem of [1] holds without the commutativity.

• East Asian mathematical journal
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• v.24 no.2
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• pp.187-190
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• 2008

As a generalization of BCI-algebras, the notion of pseudo-BCI algebras is introduced, and some of their properties are investigated. Characterizations of pseudo-BCI algebras are established. Some conditions for a pseudo-BCI algebra to be a pseudo-BCK algebra are given.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.171-181
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• 2012

The notion of implicative vague ideals of BCK-algebras is defined, and several properties of it are investigated. Relations between a vague ideal and an implicative vague ideal is discussed. Characterizations of an implicative vague ideal are considered.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.399-408
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• 2009

A bipolar fuzzy translation and a bipolar fuzzy S-extension of a bipolar fuzzy subalgebra in a BCK/BCI-algebra are introduced, and related properties are investigated.

• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.61-65
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• 1995

In this paper we introduce the notion of Artinian and obtain some properties of Artinian and Noetherian BCK-algebras.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1373-1386
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• 2017

In this paper, we introduce and explore suggested notions of 'above', 'below' and 'between' in general groupoids, Bin(X), as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), d/BCK-algebras, linear groupoids over fields and special cases, in order to illustrate the usefulness of these ideas. Additionally, for groupoid-classes (e.g., BCK-algebras) where these notions have already been accepted in a standard form, we look at connections between the several definitions which result from our introduction of these ideas as presented in this paper.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.7-13
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• 2012

In this paper we consider pseudo-BCK/BCI-algebras. In particular, we consider properties of minimal elements ($x{\leq}a$ implies x = a) in terms of the binary relation $\leq$ which is reflexive and anti-symmetric along with several more complicated conditions. Some of the properties of minimal elements obtained bear resemblance to properties of B-algebras in case the algebraic operations $\ast$ and $\circ$ are identical, including the property $0{\circ}(0{\ast}a)$ = a. The condition $0{\ast}(0{\circ}x)=0{\circ}(0{\ast}x)=x$ all $x{\in}X$ defines the class of p-semisimple pseudo-BCK/BCI-algebras($0{\leq}x$ implies x = 0) as an interesting subclass whose further properties are also investigated below.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.377-402
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• 2007

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.

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.353-360
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• 2009

Fuzzy translations, (normalized, maximal) fuzzy extensions and fuzzy multiplications of fuzzy subalgebras in BCK/BCI-algebras are discussed. Relations among fuzzy translations, (normalized, maximal) fuzzy extensions and fuzzy multiplications are investigated.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.461-470
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• 2006

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 $\={A}$ in X, we construct a fuzzy implicative ideal of X containing $\={A}$.