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Zbl 1069.06012
Jun, Young Bae; Xin, Xiaolong; Roh, Eun Hwan
The role of atoms in BCI-algebras.
(English)
[J] Soochow J. Math. 30, No. 4, 491-506 (2004). ISSN 0250-3255

The authors define some kinds of atoms in BCI-algebras and consider their properties. They show that any finite BCI-algebra $X$ is generated by $I$-atoms which are elements $a\ne 0$ such that if $x\le a$ then $x= a$ for every $x\in X -\{0\}$. They also find a condition for a BCI-algebra to be a proper $I$-branch BCI-algebra, that is: Theorem 3.26. If a BCI-algebra $X$ satisfies the following conditions: (1) $c* a= c$ for all $a\in L_K(X)$ and $c\in V(a) -\{a\}$, (2) every subalgebra $S$ of $X$ with $|S|\ge 3$ is an ideal of $X$, then $X$ is a proper $I$-branch BCI-algebra, where \align X_+ &= \{a\in X\mid 0\le a\},\\ L_K(x) &= \{a\in X_+- \{0\}\mid x\le a\Rightarrow x= a\ (\forall x\in X- \{0\})\},\\ V(a) &= \{x\in X\mid a\le x\}.\endalign
[Michiro Kondo (Inzai)]
MSC 2000:
*06F35 BCK-algebras, etc.

Keywords: atoms; BCI-algebras

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