• 대한수학회논문집
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• 제9권2호
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• pp.265-270
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• 1994

In 1966, Iseki [2] introduced the notion of BCI-algebras which is a generalization of BCK-algebras. Lei and Xi [3] discussed a new class of BCI-algebra, which is called a p-semisimple BCI-algebra. For p-semisimple BCI-algebras, a subalgebra is an ideal. But a subalgebra of an arbitrary BCI/BCK-algebra is not necessarily an ideal. In this note, a BCI/BCK-algebra that every subalgebra is an ideal is called a normal BCI/BCK-algebra, and we give characterizations of normal BCI/BCK-algebras. Moreover we give a positive answer to the problem which is posed in [4].(omitted)

• 호남수학학술지
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• 제29권2호
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• pp.289-297
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• 2007

In this paper, we show that BCI-algebras P and Q are injective if and only if its direct sum P $\oplus$ Q is injective. Moreover, we obtain the equivalent conditions for a p-semisimple BCI-algebra to be p-injective.

• 호남수학학술지
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• 제31권4호
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• pp.601-609
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• 2009

The notion of generalized derivations of a BCI-algebra is introduced, and some related properties are investigated. Also, the concept of a torsion free BCI-algebra is introduced and some properties are discussed.

• Journal of applied mathematics & informatics
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• 제38권1_2호
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• pp.65-77
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• 2020

In this paper, for any non-empty subsets A, I of a pseudo BCI-algebra X, we introduce the concept of pseudo p-closure of A with respect to I, denoted by A^pc_I, and investigate some related properties. Applying this concept, we state a necessary and sufficient condition for a pseudo BCI-algebra 1) to be a p-semisimple pseudo BCI-algebra; 2) to be a pseudo BCK-algebra. Moreover, we show that A^pc_{0} is the least positive pseudo ideal of X containing A, and characterize it by the union of some branches. We also show that the set of all pseudo ideals of X which A^pc_I = A, is a complete lattice. Finally, we prove that this notion can be used to define a closure operation.

• 대한수학회논문집
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• 제27권1호
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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.

• 충청수학회지
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• 제12권1호
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• pp.33-41
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• 1999

In this paper, we introduce a new notion, called an QS-algebra, which is related to the areas of BCI/BCK-algebras and discuss the G-part of QS-algebras.