The relation $\leq$ and $<$ on a Boolean algebra are defined as:

$x\leq y$ if and only if $x\vee y=y$

$x<y$ means $x\leq y$ but $x\neq y$

$x\geq y$ means $y\leq x$

$x>y$ means $y<x$

Considering the above definitions, which of the following is not true in the Boolean algebra?

$(i)$ If $x\leq y$ and $y\leq z$, then $x\leq z$
$(ii)$ If $x\leq y$ and $y\leq x$, then $x=y$
$(iii)$ If $x<y$ and $y<z$, then $x\leq y$
$(iv)$ If $x<y$ and $y<z$, then $x<y$