IEEE 754 floating point numbers contain the concept of (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return if either argument is ). That is, is an absorbing element. However, from an algebraic point of view, FP numbers are very ugly, as they are not even associative.
So instead, consider the following construction: Let $(K, +_{K}, ⋅_{K})$ be any field. Extend $K$ by a singular element $K' ≔ K∪\{\}$, and augment the field operations as follows:
$$\begin{aligned} x +_{K'} y &≔ \begin{cases} :& ∈\{x,y\} \\ x +_{K} y :& \text{else}\end{cases} & x ⋅_{K'} y &≔ \begin{cases}\hfill:& ∈\{x,y\} \\ x ⋅_{K} y\hfill:& \text{else}\end{cases} \end{aligned}$$
The resulting algebraic structure $(K', +_{K'}, ⋅_{K'})$ still satisfies all field axioms except for additive and multiplicative inverses, which fail for the singular distinguished element .
Question: Does this algebraic structure have a name? Is it studied in the literature?
The concept seems related to that of the Maybe/Option-Monad, but I am not knowledgeable in category theory.
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