Countable ordered groups and Weihrauch reducibility

This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem [11] on the order types of countable ordered groups. Solomon [14] showed that the theorem is equivalent to ${\Pi }_1^1$-$C{A}_0$, the strongest of the big five subsystems of second order arithmetic. We show that the strength of the theorem comes from having a dense linear order without endpoints in its order type. Then, we show that for the related Weihrauch problem to be strong enough to be equivalent to $\widehat{\mathrm{WF}}$ (the analog problem of ${\Pi }_1^1$-$C{A}_0$), an order-preserving function is necessary in the output. Without the order-preserving function, the problems are very much to the side compared to analog problems of the big five.