is said to be [*regular*]{} \[DLM2\] if any weak $V$-module is a direct sum of irreducible (ordinary) $V$-modules. If $V$ is regular, it was proved in \[DLM2\] that $V$ has only finitely many inequivalent irreducible modules. Examples of regular vertex operator algebras were given in \[DLM2\].

It is well known (cf. \[B\], \[FFR\], \[Li1\], \[Li4\], \[MP\]) that there is a Lie algebra $g(V)$ associated to $V$. More precisely, $$\begin{aligned}
g(V)=\hat{V}/d\hat{V},\end{aligned}$$ where $$\hat{V}=V\otimes {\C}[t,t^{-1}]\;\;\mbox{and } 
d=L(-1)\otimes 1+1\otimes {d\over dt},$$ with the following Lie bracket: $$\begin{aligned}
\label{ecomm}
[u(m),v(n)]=\sum_{i\ge 0}{m\choose i}(u_{i}v)(m+n-i)\end{aligned}$$ for $u,v\in V,\; m,n\in {\Z}$, where $u(m)=u\otimes t^{m}+d\hat{V}$. Furthermore, $g(V)$ is a ${\Z}$-graded Lie algebra where $\deg u(m)=\wt u-m-1$ for homogeneous $u\in V$ and for $m\in {\Z}$. Let $W$ be a $g(V)$-module. A vector $w$ of $W$ is said to be [*restricted*]{} if for every $v\in V$, $v(m)w=0$ for $m$ sufficiently large. Denote by $W^{res}$ the space of all the restricted vectors of $W$. Since for any $u,v\in V$, $u_{i}v\ne 0$ for only finitely many nonnegative integer $i$, it follows from (\[ecomm\]) that $W^{res}$ is a submodule of $W$. A $g(V)$-module $W$ is said to be [*restricted*]{} if $W=W^{res}$. In general, it is clear that $W^{res}$ is the unique maximal restricted submodule 