• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.51-62
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• 2002

The present paper gives a new construction of a quotient BCI(BCK)-algebra X/${\mu}$ by a fuzzy ideal ${\mu}$ in X and establishes the Fuzzy Homomorphism Fundamental Theorem. We show that if ${\mu}$ is a fuzzy ideal (closed fuzzy ideal) of X, then X/${\mu}$ is a commutative (resp. positive implicative, implicative) BCK(BCI)-algebra if and only if It is a fuzzy commutative (resp. positive implicative, implicative) ideal of X Moreover we prove that a fuzzy ideal of a BCI-algebra is closed if and only if it is a fuzzy subalgebra of X We show that if the period of every element in a BCI-algebra X is finite, then any fuzzy ideal of X is closed. Especiatly, in a well (resp. finite, associative, quasi-associative, simple) BCI-algebra, any fuzzy ideal must be closed.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.433-436
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• 2006

In this note we show that a representable difference algebra is equivalent to a commutative BCK-algebra.

• Journal for History of Mathematics
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• v.20 no.1
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• pp.73-82
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• 2007

This paper delineates some history and developments of Wajsberg hoops, MV-algebras and commutative BCK-algebras. It also discuss some relations on them. Finally, it shows that every Wajsberg hoop is a commutative BCK-algebra.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.261-262
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• 2016

We present a short and very elementary proof that each finite BCK-algebra is solvable. It is a positive answer to the question posed in [4].

• The Pure and Applied Mathematics
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• v.15 no.1
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• pp.73-84
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• 2008

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 commutative ideals in BCK-algebras. The notion of an intuitionistic fuzzy commutative ideal of a BCK-algebra is introduced, and some related properties are investigated. Characterizations of an intuitionistic fuzzy commutative ideal are given. Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy commutative ideal are given. Using a collection of commutative ideals, intuitionistic fuzzy commutative ideals are established.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.183-196
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• 2011

Molodtsov [5] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to an implicative ideal of BCK-algebras. The notion of implicative soft ideals in BCK-algebras and implicative idealistic soft BCK-algebras is introduced, and related properties are investigated. Relations between implicative soft ideals and commutative (resp. positive implicative) soft ideals are discussed. Also, relations between implicative idealistic soft BCK-algebras and commutative (resp. positive implicative) idealistic soft BCK-algebras are provided.

• Bulletin of the Korean Mathematical Society
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• v.46 no.1
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• pp.147-153
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• 2009

In this paper, we give some constructions of implicative/commutative d-algebras which are not BCK-algebras. This demonstrate that the notion of implicative/commutative d-algebras are indeed generalizations of the same in BCK-algebras.

• Journal of the Chungcheong Mathematical Society
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• v.12 no.1
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• pp.15-21
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• 1999

In this paper, we introduce the notion of anti fuzzy prime ideals in a commutative BCK-algebra and obtain some properties of it.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.707-720
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• 2008

Molodtsov [12] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to commutative ideals of BCK-algebras, The notions of commutative soft ideals and commutative idealistic soft BCK-algebras are introduced, and their basic properties are investigated. Examples to show that there is no relations between positive implicative idealistic soft BCK-algebras and commutative idealistic soft BCK-algebras are provided.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.4
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• pp.395-404
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• 2020

Based on a sub-BCK-algebra K of a BCI-algebra X, the notions of fuzzy (K, ε)-subalgebras, fuzzy (K, ε)-ideals and fuzzy commutative (K, ε)-ideals are introduced, and their relations/properties are investigated. Conditions for a fuzzy subalgebra/ideal to be a fuzzy (K, ε)-subalgebra/ideal are provided.