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? )