  We fix $z_0\in\mathbb C$ and a field $\mathbb F$ with $\mathbb C\subset
\mathbb F \subset \mathcal M_{z_0}:=$ the field of germs of meromorphic
functions at $z_0$. We fix $f_1,\ldots,f_r\in \mathcal M_{z_0}$ and we consider
the $\mathbb F$-algebras $S:=\mathbb F[f_1,\ldots,f_r]$ and $\overline
S:=\mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]$. We present the general
properties of the semigroup rings \begin{align*} & S^{hol}:=\mathbb
F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r
\text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\ & \overline
S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}:
(a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic
at }z_0], \end{align*} and we tackle in detail the case in which $\mathbb
F=\mathcal M_{<1}$ is the field of meromorphic functions of order $<1$ and
$f_j$'s are meromorphic functions over $\mathbb C$ of finite order with a
finite number of zeros and poles.
