Fields involving NaN

Motivation

NaN as a value exists in IEEE floating-point numbers. Since every operation involving NaN has NaN as the outcome, IEEE floating-point numbers are not fields. I want to define a new algebraic structure so NaN could be encapsulated.

Definitions

A field $(F,+,\times)$ is NaNable if there exists a binary operation $\sim$ such that $(F \cup \{\text{NaN}\}, \sim, +)$ is a field, where $\text{NaN}$ is the identity element of $\sim$.

$(F \cup \{\text{NaN}\}, \sim, +,\times)$ is a NaNification of $(F,+,\times)$, and is a NaN-field.

Examples

Examples of NaNable finite fields include $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_7$, and $\mathbb{Z}_{31}$. The operation tables of NaNification of $\mathbb{Z}_3$ is:

$$ \begin{array}{c|cccc} \sim & \text{NaN} & 0 & 1 & 2 \\ \hline \text{NaN} & \text{NaN} & 0 & 1 & 2 \\ 0 & 0 & \text{NaN} & 2 & 1 \\ 1 & 1 & 2 & \text{NaN} & 0 \\ 2 & 2 & 1 & 0 & \text{NaN} \\ \end{array} $$

$$ \begin{array}{c|cccc} + & \text{NaN} & 0 & 1 & 2 \\ \hline \text{NaN} & \text{NaN} & \text{NaN} & \text{NaN} & \text{NaN} \\ 0 & \text{NaN} & 0 & 1 & 2 \\ 1 & \text{NaN} & 1 & 2 & 0 \\ 2 & \text{NaN} & 2 & 0 & 1 \\ \end{array} $$ $$ \begin{array}{c|ccc} \times & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 1 \\ \end{array} $$

Note that multiplication involving $\text{NaN}$ is not defined yet.

Questions

1. Is there a notable example of NaNable infinite field?

2. Is there a dedicated name for $\sim$? Not as a symbol, but as an operation?

3. It turns out that if every multiplication involving $\text{NaN}$ is defined as $\text{NaN}$, distributivity is satisfied in the NaN-field. Does this mean I should define it accordingly?

from Hot Weekly Questions - Mathematics Stack Exchange