The most elementary math operation always has two operands and an operator. For instance, the addition of two operands $a$ and $b$ can be represented as:
$$a + b, \quad a, b \in \mathbb{C}$$
This notation allows for the selection of any complex number $a$ and $b$ to perform addition but there is no flexibility in selection of the operator. Assume there is a set $\Omega$ that encompasses all possible operators ($+$, $-$, $\times$, $\div$, ^, √, %, $\dots$) so that we can achieve the form:
$$a\, O\, b, \quad O \in \Omega \text{ where } \Omega \text{ is the set of all possible arithmetic operators}$$
The idea behind using this form is that we can then define a set of infinite results obtained by performing a different combination of operator and second operand on a given number $x \in \mathbb{C}$ like so:
$$R_x = \{x \, O \, y \,|\, O \in \Omega, y \in \mathbb{C}\}$$
Can such a set $\Omega$ be precisely defined so that it encompasses every arithmetic operator that can be used between two operands? If this is possible, how can $\Omega$ be defined? If it cannot be defined, what are the limitations?