Let $\mathbb{Q}^*$ be the set of finite sequences of rationals, and let $x \sim y$ if and only if

they have the same length, and for all $1 \le i,j \le length(x)$, $x_i \lt x_j$ iff $y_i \lt y_j$ ('order equivalence')

if $x_i \in \mathbb{Z}^+$ then $y_i \in \mathbb{Z}^+$

if $x_i \not\in \mathbb{Z}^+$ then $x_i = y_i$.

**Definition:** We say that the function $T:\mathbb{Q}^* \to \mathbb{Q}^*$ is a *uniform transformation* (with respect to $\sim$) if and only if for all $x \in \mathbb{Q}^*$ we have $(x,Tx) \sim (Tx,TTx)$.

We also have analogous definitions when restricting attention to $\mathbb{Q}^k$ (note that a uniform transformation necessarily restricts to a function $\mathbb{Q}^k \to \mathbb{Q}^k$ for all $k$).

This definition is due to Harvey Friedman (http://www.cs.nyu.edu/pipermail/fom/2012-March/016316.html), and I can only guess that the latter condition involving $(x,Tx)$ involves concatenation of strings [EDIT: it cannot be concatenation, see Aaron's comment below. If anyone can reverse engineer a definition, that would be great. I'll see if I can get Friedman to clarify]. Friedman arrived at this definition by abstracting the required properties of a function he calls $\mathbb{Z}^+\!\!\uparrow$ - it takes a finite string, finds the first entry which is not a positive integer, and then adds 1 to all entries after that. For example.

$$\mathbb{Z}^+\!\!\uparrow(1,3/2,3,5) = (1,3/2,4,6).$$

[EDIT 2: my definition of $\mathbb{Z}^+\!\!\uparrow$ is wrong, see Aaron's answer. The correct definition is to add 1 to all (necessarily integer) entries larger than the supremum of the non-integer entries.]

Now I was wondering if there are any other examples of uniform transformations. I hope there are, because $\mathbb{Z}^+\!\!\uparrow$ looks to me fairly contrived (of course, it is contrived), and others Friedman has asked personally have said the same. On the other hand, perhaps this function (or one like it) arises naturally from some combinatorial problem. My questions

A. What are other examples of uniform transformations?

B. Can we arrive at the function $\mathbb{Z}^+\!\!\uparrow$ 'naturally'?

Any other thoughts are appreciated.

uniform transformationimplies that $x \sim Tx.$ But for $T=\mathbb{Z}^+\!\!\uparrow$ and $x= 3/2 ,1$ we have $y=Tx=3/2,2$ so order equivalence fails: $x_1 \gt x_2$ but $y_1 \lt y_2.$ My understanding is that Harvey Friedman goes to great effort to make his examples as natural seeming as possible. $\endgroup$ – Aaron Meyerowitz Mar 15 '12 at 17:26