I am reading Friedman & Stanley *A Borel reducibility theory for classes of countable structures* (J. Symbolic Logic 54 (1989), 894–914; [MR1011177](http://www.ams.org/mathscinet-getitem?mr=1011177)) and a caret (${}^\wedge$) appears as notation in the following contexts without definition:


1. $\mathbb{Q} \times (-\infty^\wedge \mathbb{Q})$. The type of $(-\infty^\wedge \mathbb{Q})$ should be a linear order.
2. $\sigma^\wedge m$, where $\sigma \in \omega^n$ and $m \in \omega$. The type of the result should be an element of $\omega^{n+1}$.
3. $ \mathbb{Q}^\wedge k + 2^\wedge \mathbb{Q}$ where $k$ is an integer. The type of the result should be a linear order.


Context on (1) suggests that $-\infty^\wedge \mathbb{Q}$ should be the linear order obtained by adding a bottom element to $\mathbb{Q}$. The intended meaning of (2) is clear. I would like to know what (3) means! Thanks.



Meaning of notation $\mathbb{Q}^\wedge k$, $-\infty^\wedge \mathbb{Q}$ for linear orders
It looks to me like a primitive concatenation symbol, like what might today be denoted $\sigma^\frown\eta$, and this would align with your expectation in the last paragraph. If we regard $k$ as the $k$ point order (as it is with the von Neumann ordinals), then $\mathbb{Q}$^$k$ would mean adding k points on top of the rational order, and $2$^$\mathbb{Q}$ would mean adding two points at the bottom of the rational order. 


