LAUSR

Abstract We fix z0 ∈ ℂ and a field ???? with ℂ ⊂ ???? ⊂ ????z0 := the field of germs of meromorphic functions at z0. We fix f1, …, fr ∈ ????z0 and we consider the ????-algebras S := ????[f1, …, fr] and S¯:=F[f1±1,…,fr±1]. $\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings Shol:=F[fa:=f1a1⋯frar:(a1,…,ar)∈Nr and fa is holomorphic at z0],S¯hol:=F[fa:=f1a1⋯frar:(a1,…,ar)∈Zr and fa is holomorphic at z0], $$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$ and we tackle in detail the case ???? = ????<1, the field of meromorphic functions of order < 1, and fj’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.