The answer to the first question is yes and the answer to the second question is no. As Ovidiu Costin confirmed in an email to me, the desired isomorphism can be constructed using an idea I learned from him regarding how to define sin/cos on all the surreals. The idea in Ovidiu's words follows, where N ranges over the omnific integers (finite and infinite).

With sin/cos the idea is not mine but Martin's (or it even goes back to Conway). What it gives is the following prescription: sin(2 pi N delta)=sin(delta), if delta\in [0,2\pi). This can be taken as a definition as well. Similarly with cos. Clearly sin/cos are well defined on all surreals. Any isomorphism should now be straightforward.

Regards,

Philip Ehrlich

The answer to the first question is yes and the answer to the second question is no. As Ovidiu Costin confirmed in an email to me, the desired isomorphism can be constructed using an idea I learned from him regarding how to define sin/cos on all the surreals. The idea in Ovidiu's words follows, where N ranges over the omnific integers (finite and infinite).

With sin/cos the idea is not mine but Martin's (or it even goes back to Conway). What it gives is the following prescription: sin(2 pi N+delta)=sin(delta), if delta\in [0,2\pi). This can be taken as a definition as well. Similarly with cos. Clearly sin/cos are well defined on all surreals. Any isomorphism should now be straightforward.

Regards,

Philip Ehrlich