• Honam Mathematical Journal
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• v.32 no.2
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• pp.217-226
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• 2010

Using the Bu$\c{s}$neag's model ([1, 2, 3]), the notion of pseudo-valuations (valuations) on a ${\mathbf{BCK/BCI}}$-algebra is introduced, and a pseudo-metric is induced by a pseudo-valuation on ${\mathbf{BCK/BCI}}$-algebras. Based on the notion of (pseudo) valuation, we show that the binary operation in ${\mathbf{BCK/BCI}}$-algebras is uniformly continuous.

• Bulletin of the Korean Mathematical Society
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• v.34 no.2
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• pp.199-204
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• 1997

To develope the theory of BCI-algebras, the idel theory plays an important role. The first author [4] introduced the notion of T-ideal in BCI-algebras. In this paper, we first construct a special set, called T-part, in a BCI-algebra X. We show that the T-part of X is a subalgebra of X. We give equivalent conditions that the T-part of X is an ideal. By using T-part, we provide an equivalent condition that every ideal is a T-ideal.

• East Asian mathematical journal
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• v.21 no.2
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• pp.171-181
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• 2005

In this paper we investigate some more properties of of $S_3$-algebras. We also prove that the class of $S_3$-algebras is contained in the class of commutative BCI-algebras.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.11-24
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• 2013

The notion of intersectional soft sets is introduced, and several examples are given. The application of intersectional soft sets to BCK/BCI-algebras is discussed.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.531-538
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• 1994

This paper is a continuation of [1] and [3]. In [3], the notion of BCI-G parts of BCI-algebras was introduced and various properties were investigated. In this paper, we consider the inverse of [3; Theorem 15], and define a KG-union BCI-algebra and investigate their properties.

• 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.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.859-870
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• 2015

The notion of quasi-valuation maps based on a subalgebra and an ideal in BCK/BCI-algebras is introduced, and then several properties are investigated. Relations between a quasi-valuation map based on a subalgebra and a quasi-valuation map based on an ideal is established. In a BCI-algebra, a condition for a quasi-valuation map based on an ideal to be a quasi-valuation map based on a subalgebra is provided, and conditions for a real-valued function on a BCK/BCI-algebra to be a quasi-valuation map based on an ideal are discussed. Using the notion of a quasi-valuation map based on an ideal, (pseudo) metric spaces are constructed, and we show that the binary operation * in BCK-algebras is uniformly continuous.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.499-505
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• 1997

This paper is a continuation of [3]. We prove that if A is quasi-associative (resp. an implicative) ideal of a BCI-algebra X then the k-nil radical of A is a quasi-associative (resp. an implicative) ideal of X. We also construct the quotient algebra $X/[Z;k]$ of a BCI-algebra X by the k-nhil radical [A;k], and show that if A and B are closed ideals of BCI-algebras X and Y respectively, then

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.601-611
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• 1996

In this paper, we introduce the notion of weakly associative BCI-algebras and investigate structure of it. Some of characterizations of elements of the quasi-associative part Q(X) of a BCI-algebra X are shown.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.65-74
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• 2008

The notion of vague quick ideals of BCK/BCI-algebras is introduced, and several properties are investigated. Relations between a vague ideal, a vague BCK/BCI-algebra and a vague quick ideal are provided. A condition for a vague quick ideal to be a vague ideal is given.