Star as Convergence in a Kleene Algebra In a Quantale, or ``Standard Kleene Algebra'', one uses as definition of star to be the sum of the powers of the given element and this is well-defined since arbitrary sums converge in such settings. That is,
$$ a^* := \sum_{i \in \mathbb{N}} a^i  \\\qquad\qquad\qquad \qquad = 1 + a + a^2 + a^3 + a^4 + \cdots$$
Can it be proven that this sum holds in any Kleene Algebra? 
That is, can one prove in an arbitrary Kleene Algebra that (1) the sum converges and (2) it does so to $a^*$?
Thank-you :-)
( Unrealted: Why is there no Kleene-Algebra tag? )
 A: If
$$
a^*=\sum_{i\in\mathbb{N}} a^i
$$
holds within a Kleene algebra $K$, then $K$ is said to be $^*$-continuous. 
There are Kleene algebras which are not $^*$-continuous. One such Kleene algebra is given in "On Kleene Algebras and Closed Semirings" (Kozen 1990). Here's the construction: $K=\mathbb{N}\times\mathbb{N}\cup\{\top,\bot\}$ and $\cdot$ is given for $k\in K$ and $(a,b),(c,d)\in \mathbb{N}\times\mathbb{N}$
 by
$$
\begin{align}
k\cdot \bot &= \bot \cdot k =\bot\\
k\cdot\top &=\top\cdot k=\top, k\neq \bot\\
(a,b)\cdot (c,d)&=(a+c,b+d)
\end{align}
$$
where $+$ is the normal addition on natural numbers. The $+$ in $K$ is defined as
$$
(a,b)+(c,d)=
\begin{cases}
(c,d) &: a\leq c~\text{ or }a=c, ~b\leq d\\
(a,b) &: \text{otherwise}
\end{cases}
$$
and $\top+k=\top$ for all $k\in K$ and $\bot + k=k$ for all $k\in K$. Notice that $+$ then is the least upper bound with respect to lexicographical ordering. Finally, $^*$ is defined as
$$
k^*=\begin{cases}
(0,0) &:k=\bot~\text{ or }~k=(0,0)\\
\top &:\text{otherwise}.
\end{cases}
$$
Let $0=\bot$ and $1=(0,0)$. Then it is straightforward to check that $(K, +, \cdot, ^*, 0, 1)$ is a Kleene algebra. It's not $^*$-continuous because $(0,1)^*=\top$ but
$$
\begin{align}
\sum_{n\in\mathbb{N}} (0,1)^n &=\sup_{n\in\mathbb{N}}~(0,1)^n\\
&=\sup_{n\in\mathbb{N}}~(0,n)\\
&=(1,0).
\end{align}
$$
