Is it possible to define an extension of the probability semiring $(\mathbb{R}^+, +, \times, 0, 1)$ such that
- Closure $a^* = 1 + a + a^2 + \ldots$ is defined for every element of the semiring, not just for those less than $1$;
- The semiring is divisible and cancellable.
- There exists a total order on the elements of the semiring, such that
- $a^* > b^*$ for any $a > b$
- Higher-order infinities are greater than the lower-order ones: $(a^*)^* > b^*$ for any $a$ and $b$ (except where the corresponding geometric series converges to a finite value).
- A sum or a product of a finite number of infinities is less than the next level of infinity: $a^*b^*c^* < (d^*)^*$ (again, assuming that all the involved geometric series diverge).
- [optional] $0 < \frac{1^*}{1^* + 2^*} < \varepsilon$ for any $\varepsilon \in \mathbb{R}_{>0}$ (so, division by a higher-order infinity results in an "infinitesimal" value). Analogously, $1 < \frac{2^*}{1^* + 2^*} < 1+\varepsilon$ for any $\varepsilon \in \mathbb{R}_{>0}$.
- The semiring can be efficiently implemented on a computer, meaning that the representation of the semiring elements is finite and, preferrably, short (assuming the floating point approximation being used), and the operations can be implemented with a reasonably low space and time complexity.
Are any such extensions known in literature? If not, is there a fundamental reason why such a semiring cannot be consistently defined?
To provide some background, I'm wondering whether it's possible to define a semiring such that every weighted automaton over that semiring will have well-defined values and a normalizer despite the presence of "diverging" loops.