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The role of atoms in BCI-algebras. (English) Zbl 1069.06012

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 $x\in X-\left\{0\right\}$. 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 $a\in {L}_{K}\left(X\right)$ and $c\in V\left(a\right)-\left\{a\right\}$,

(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

$\begin{array}{cc}\hfill {X}_{+}& =\left\{a\in X\mid 0\le a\right\},\hfill \\ \hfill {L}_{K}\left(x\right)& =\left\{a\in {X}_{+}-\left\{0\right\}\mid x\le a⇒x=a\phantom{\rule{4pt}{0ex}}\left(\forall x\in X-\left\{0\right\}\right)\right\},\hfill \\ \hfill V\left(a\right)& =\left\{x\in X\mid a\le x\right\}·\hfill \end{array}$

##### MSC:
 06F35 BCK-algebras, BCI-algebras
##### Keywords:
atoms; BCI-algebras