I am trying to design a particular notation, described as follows:
$X$ is a finite set of elements.
$s_k, k = 1...n, n < 10$ are few, different, mapping function.
Each $s_k$ function maps elements of $X$ into 3 disjoint sets: $U$, $D$ and $I$ .
For each function $s_k$, if an element $x$ is put in $U$, $s_k(x) = 1$; if it is put in $D$, $s_k(x) = -1$; finally, if it is put in $I$, $s_k(x) = 0$.
The $s_k$ functions are designed in such a way that if each element can be mapped to $1$ by at most one function, and can be mapped to $-1$ by at most another (NOT by the same) function. However, each element can be mapped to $0$ even if mapped to $1$, $-1$ or $0$ by another function (but NOT by the same function).
Is there a particular notation for such case?
Thanks
In the case of 2 disjoint sets $A,B$, which together form a partition of their disjoint union $Y=A \cup B$, one speaks about the characteristic function of the subset $A$ as the function $\chi : Y \to \{0,1\}$ defined by $\chi(x)=1$ for $x \in A$ and $\chi(x)=0$ for $x \in B=X-A$.
Given 3 disjoint sets $U,D,I$ which together partition their disjoint union $Y = U \cup D \cup I$, you could perhaps speak about a 3-fold characteristic function $\chi:Y \to \{-1,0,1\}$ defined by $$\chi(x)= \begin{cases} 1 & \text{if $x \in U$} \\ 0 & \text{if $x \in D$} \\ -1 &\text{if $x \in I$} \end{cases} $$