algebraic structure consisting of two fields with common multiplication operation

I am looking for properties of an algebraic structure on a set defined as follows: - two fields with distinct operations of addition (thus two distinct additive zeros) and a common operation of multiplication.

An example would be the set $\mathbb{R}$ of real numbers and two operations of addition defined on it. For $a,b\in \mathbb{R}$:

• one is the usual addition $a+b$;
• the other is $a\dot{+}b=\frac{1}{\frac{1}{a}+\frac{1}{b}}$.

The inverse of $a\in \mathbb{R}$ under both operations is $-a$. The additive zero of $+$ operation is the $0$, the additive zero of $\dot{+}$ is $\infty$ if we allow $\infty=\frac{1}{0}$.

Any mention, examples, thoughts would be very welcome.

• This is not clear. Can you give a detailed example of the structure you have in mind? – lulu Jul 26 at 12:13
• In particular, how could there be two different additive zeroes? Since $0_i\times x=0_i$ for all $x$ in the field we see that $0_1\times 0_2=0_2\times 0_1$ is both $0_1$ and $0_2$. – lulu Jul 26 at 12:20
• Not following. As a general rule, it's a bad idea to talk about algebraic structures with no examples in mind. The formalism is meant to generalize the properties you observe in particular examples. Here, I don't think you have any actual examples in mind. If you do, please edit your post to describe one or two examples in detail. – lulu Jul 26 at 12:36
• Again, not following. If you have a precise example, please edit your post to include a detailed discussion of it. – lulu Jul 26 at 12:49
• In the example you don't have two field structures on the same set. The joint set is $S=\Bbb{R}\cup\{\infty\}$. You have one field structure on the set $S\setminus\{\infty\}$ and another on $S\setminus\{0\}$. They happen to share the multiplicative group, but the zero of one is outside the other field altogether. – Jyrki Lahtonen Jul 26 at 20:24