order theory – How to define a function that has these specific properties?


Suppose $x = (x_1,x_2,dots,x_K) in mathbb{Z}^K_{geq 0}$. For $x,y in mathbb{Z}^K_{geq 0}$, we write $x succ y$ or $y prec x$ if $x neq y$ and

begin{align*}
x_{i(x,y)} > y_{i(x,y)},quad text{ where } quad i(x,y) := max{ i: x_i neq y_i}.
end{align*}

That is, for any two vectors $x$ and $y$ that are not equal, we let $i(x,y)$ be the last position on which they differ and say that $x succ y$ if the coordinate of $x$ at $i(x,y)$ is larger than the corresponding coordinate of $y$. We write $x succeq y$ if either $x = y$ or $x succ y$, and similarly for $x preceq y$. This is a total order.

For example, if $x = (7,2,1,0,0)$ and $y = (6,3,1,0,0)$ then $y succ x$ because they are equal on the last three positions and the next position that they differ is the second coordinate, since 3>2 we conclude that $y succ x$. This is called “reflected lexicographic order”.

Now, let $mx(x) = max{k: x_k > 0}$, we are interested in defining a function $f: mathbb{Z}^K_{geq 0} rightarrow (0,K+1)$ that has the following properties:

  • $f(0,0,ldots,0) = 0$
  • $mx(x) leq f(x) < mx(x)+1$ (Note that when one of the coordinates of x is 1 and the rest are 0, then $f(x)= mx(x)$, for example let $x = (0,1,0,0,0)$, then $f(x)=mx(x)=2$)
  • $f(.)$ is strictly increasing on $mathbb{Z}^K_{geq 0}$ wrt. the total-ordering defined above
  • The effect of adding a positive value to coordinate $k$ should be smaller than adding the same value to coordinate $k+1,….,K$, having all the other values fixed, sth like convexity property but I’m not sure if the exact definition of convexity applies here.
    For example suppose $K=5$, $f(0,3,0,0,0) – f(0,2,0,0,0) leq f(0,0,3,0,0) – f(0,0,2,0,0)$

I could define a function that has the first three properties, but not the fourth one:
For any $x in mathbb{Z}^K_{geq 0}$, let $g_{k}(x) = prod_{i=k}^{K} (1+i)^{-x_i}$ for $k=2,dots,K$ and $g_{K+1}(x) = 1$.
begin{align}
f(x) := sum_{k=2}^{K+1} k g_k(x) big(1 – k^{-x_{k-1}}big).
end{align}

$f(0,3,0,0,0) – f(0,2,0,0,0) = 2.888889 – 2.666667 = 0.222222$
but $f(0,0,3,0,0) – f(0,0,2,0,0) = 3.9375 – 3.75 = 0.1875$

How to define $f(.)$ so that it follows all the 4 properties?

PS: This is cross-post from Math.SE (I flagged it there to be migrated to mathoverflow but no one has migrated it)