# 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) Posted on Categories Articles