• Journal of the Korean Institute of Intelligent Systems
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• v.12 no.2
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• pp.182-186
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• 2002

We investigate the quotient structure of a product BCK-algebra and fundamental homomorphism theorem and show their some properties.

• The Mathematical Education
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• v.21 no.3
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• pp.13-14
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• 1983

(1) The direct product (equation omitted) $E_{I}$ of BCK-algebras $E_{I}$, (i=1, 2, 3, …, n), is a BCK-algebra. (2) Let E be a BCK-algebra and $A_1$, $A_1$, …, $A_{n}$ ideals of E. Define a mapping (equation omitted) by the rule f($\chi$)=( $A_1$$\chi$, $A_2$$\chi$, …, $A_{n}$$\chi$). Then f is a homomorphism.ism.ism.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.221-229
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• 2005

In this paper we consider the decompositions of subdirect sums and direct sums in bounded BCK-algebras. The main results are as follows. Given a bounded BCK-algebra X, if X can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ of a nonzero ideal family $\{A_i\;{\mid}\;i{\in}I\}$ of X, then I is finite, every $A_i$ is bounded, and X is embeddable in the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X is with condition (S), then it can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ if and only if it can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$, then it is isomorphic to the direct product $\prod_{i{\in}I}A_i$.