\section{Definitions}
\label{App:Definitions}

\begin{definition}\label{De:omega-cont-semiring}
An \textbf{\emph{$\omega$-continuous semiring}}
is a semiring with the following additional properties:
\begin{enumerate}
  \item 
    The relation $\sqsubseteq \,\,\, \eqdef \{(a,b) \in D \times D \mid \Exists{d} a \combine d = b \}$
    is a partial order.
  \item\label{def:wcont-supremum}
    Every $\omega$-chain $(a_i)_{i\in \bbN}$
    (i.e., for all $i \in \bbN~~a_i \sqsubseteq a_{i+1}$)
    has a supremum with respect to $\sqsubseteq$, denoted by
    $\sup_{i\in \bbN} a_i$.
  \item\label{def:wcont-inf-distributive}
    Given an arbitrary sequence $(c_i)_{i \in \bbN}$, define
    \[
      \wsum c_i \eqdef \sup \{c_0 \combine c_1 \combine \ldots \combine c_i \mid i \in \bbN\}.
    \]
    The supremum exists by (\ref{def:wcont-supremum}) above.
    Then, for every sequence $(a_i)_{i \in \bbN}$,
    for every $b \in \calS$, and every partition $(I_j)_{j \in J}$ of $\bbN$,
    the following properties all hold:
    \[
    \begin{array}{@{\hspace{0ex}}r@{\hspace{1.0ex}}c@{\hspace{1.0ex}}l@{\hspace{0ex}}}
      b \extend \left(\wsum a_i \right) & = & \wsum \left(b \extend a_i \right) \\
      \left(\wsum a_i \right) \extend b & = & \wsum \left(a_i \extend b\right) \\
      {\displaystyle \Combine\limits_{j \in J} \left( {\displaystyle \Combine\limits_{i \in I_j} a_i} \right)} & = & \wsum a_i
    \end{array}
    \]
\end{enumerate}

The notation $a^i$ denotes the $i^{th}$ term in the sequence in which
$a^0 = \srOne$ and $a^{i+1} = a^{i} \extend a$.
An $\omega$-continuous semiring has a \textbf{Kleene-star operator}
$\vphantom{a}^* \colon D \rightarrow D$ defined as follows:
$a^* = \wsum a^i$.
\end{definition}

