  Let $\omega$ be a weight on a right cancellative semigroup $S$. Let
$\|\cdot\|_{\omega}$ be the weighted norm on the weighted discrete semigroup
algebra $\ell^1(S, \omega)$. In this paper, we prove that the weight $\omega$
satisfies F-property if and only if the operator norm $\| \cdot \|_{\omega op}$
of $\| \cdot \|_{\omega}$ is exactly equal to another weighted norm $\| \cdot
\|_{\widetilde{\omega}_1}$ [Theorem 2.5 ($iii$)]. Though its proof is
elementary, the result is unexpectedly surprising. In particular, $\| \cdot
\|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, various
examples are discussed to understand the relating among $\| \cdot \|_{\omega
op}$, $\| \cdot \|_{\omega}$, and $\ell^1(S, \omega)$.
