Let $G$ be the group $G=C_3,+$ with an order relation $>$ which is unrelated to the group operation, having elements $\{0,1,2\}$ with addition modulo $3$ and let $C_2\in\{0,1\}$ with addition modulo $2$.
I can write a theorem in words: If we identify any singleton element $x_1\in G$ then the remaining two elements are an ordered pair.
Therefore I can define a function $f(x_1,x_2)\to C_2,>$ that takes two distinct elements of $C_3$ and returns $f(x_1,x_2)\in \{0,1\}$ indicating whether $x_2$ is the greater or smaller element in the order inherited on $C_3\setminus \{x_1\}$.
$f(x_1,3)=1$ for either choice of $x_1$
$f(x_1,1)=0$ for either choice of $x_1$
$f(x_1,2)\in\{0,1\}$ depending on the choice of $x_1$
But how is the domain of this function best written? None of the options I have come up with seem satisfactory. I'm struggling to express that its domain is any two distinct elements drawn from $S_3$:
$C_3\times C_2\to C_2$ seems unsatisfactory because $C_2$ in the domain is really "the ordered pair $C_3\setminus x\in C_3$, and moreover, this is just the identity function $x\in C_2\mapsto x\in C_2$
$S_3\times C_3\to C_2$ is unsatisfactory because $C_3\times C_3\to C_2$ isn't defined for $x+y:y=x$
How should I best express that the domain requires a choice of distinct elements of the same set?