Is there an 'signed' ordinal system 'stronger' than the 'original' ordinals, but 'weaker' than the surreals, with subtraction and division? Like one with negative integers and $\omega -1$ but not $0.5$, $\pi$ or any finite non-integers?
If there are some discussed, what would $\omega$ be divisible by? All integers (seems like a good idea)? And would $\omega$ be a perfect square/cube/etc in any of those systems?
And would $\{\alpha | \alpha \text{ is an 'signed' ordinal between } 0 \text{ and } \omega \text{ inclusive}\}$ be a set or a proper class?