  In this paper we show that if $Y$ is a subsemilattice of a finite semilattice
indecomposable semigroup $S$ then $|Y|\leq 2\left\lfloor
\frac{|S|-1}{4}\right\rfloor+1$. We also characterize finite semilattice
indecomposable semigroups $S$ which contains a subsemilattice $Y$ with
$|S|=4k+1$ and $|Y|=2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1=2k+1$. They
are special inverse semigroups. Our investigation is based on our new result
proved in this paper which characterize finite semilattice indecomposable
semigroups with a zero by only use the properties of its semigroup algebra.
