ON DIRECT SUMS IN BOUNDED BCK-ALGEBRAS

Title & Authors
ON DIRECT SUMS IN BOUNDED BCK-ALGEBRAS
HUANG YISHENG;

Abstract
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 $\small{\bar{\bigoplus}_{i{\in}I}A_i}$ of a nonzero ideal family $\small{\{A_i\;{\mid}\;i{\in}I\}}$ of X, then I is finite, every $\small{A_i}$ is bounded, and X is embeddable in the direct sum $\small{\bar{\bigoplus}_{i{\in}I}A_i}$ ; if X is with condition (S), then it can be decomposed as the subdirect sum $\small{\bar{\bigoplus}_{i{\in}I}A_i}$ if and only if it can be decomposed as the direct sum $\small{\bar{\bigoplus}_{i{\in}I}A_i}$ ; if X can be decomposed as the direct sum $\small{\bar{\bigoplus}_{i{\in}I}A_i}$, then it is isomorphic to the direct product $\small{\prod_{i{\in}I}A_i}$.
Keywords
bounded BCK-algebra;ideal;subdirect sum;direct sum;direct product;
Language
English
Cited by
References
1.
Z. M. Chen, Direct product theory of well BCI-algebras, J. of Fujian Teachers Univ. 3 (1987), no. 2, 17-28

2.
Z. M. Chen, Direct sums of well BCI-algebras, J. of Fujian Teachers Univ. 9 (1993), no. 2, 13-16

3.
Z. M. Chen and Y. S. Huang, Additive decomposition of BCI-algebras with condition (S), Math. Japonica 41 (1995), no. 2, 303-310

4.
K. Iseki, On BCI-algebras, Math. Seminar Notes 8 (1980), 125-130

5.
K. Iseki, On BCI-algebras with condition (S), Math. Seminar Notes 8 (1980), 171-172

6.
K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japonica 21 (1976), 351-366

7.
J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co. Seoul, Korea, 1994