Let $X$ be a BCK- algebra, $\Gamma\subset X$. Denote

$A(a)=\{\,x\in X\mid x\le a\,\}$, $T=\cup _{a\in\Gamma} A(a)$.

If for all $x\in X-T$ there exists a $x'\in \Gamma$ s.t. $x*x'\le x*a$ for all

$a\in \Gamma$, then $\Gamma$ is called a {\it root set} of the BCK-algebra\ $X$.

In this note by use of this concept  we'll prove the following theorem:

Let $\Gamma$ be a root set of a BCK-algebra\ $X=(X,*,0)$,

$T=\cup _{a\in\Gamma}A(a)$, and

$(Y,\bullet ,0)$ be a normal BCI-algebra(for its definition  see [6] or

Def. 1 below) with its BCK-part $K$ and its p-semisimple part $M$ satisfying

$X\cap Y=\{\,0\,\}$. Set $Z=X\cup Y$

Define a binary operation $\circ$ on $Z$ to be:

$x_1\circ x_2  =x_1*x_2$, if $x_1,x_2\in X$;

$x\circ y  =0$, if $x\in T, y\in \ K-\{0\}$;

$x\circ y  =x*x'$, if $x\in X-T, b\in K-\{0\} $;

$x\circ y  =0\bullet y$, if $x\in X, y\in M-\{0\} $;

$y\circ x  =y$,if $x\in X,y\in Y$;

$y_1\circ y_2 =y_1\bullet y_2$, if $y_1,y_2\in Y$,

where $x'\in\Gamma$ s.t. $x*x'\le x*a$  for all $a\in\Gamma$.

Then

$(Z,\circ,0)$  is a normal BCK-algebra and will be denoted by $X\vee _\Gamma Y$.

This work generalizes Jiang Hao's stracture [3, Theorem, 1].

Some related results will also be given. For example, An automorphism $\phi $

on $X$ has a natural extension $\tilde\phi$ on $X\vee _\Gamma Y$,

%$(X\vee _\Gamma B)\vee _\Gamma C$=

%$(X\vee _\Gamma C)\vee _\Gamma B$,

%where $C$ is a BCK-algebra with

%$X\cap B=\{\,0\,\}$, $X\cap C=\{\,0\,\}$,

%$B\cap C=\{\,0\,\}$,

etc..