I recently came across a binary operation (in a very non-algebraic context - it's a way to organize a certain updating of log-likelihood-ratios) and was idly wondering whether it is any kind of reasonable algebraic object. The answer may well be no but it does satisfy some properties that look like those of a ring.
Let $\boxplus \colon \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be given by $a \boxplus b = \log\left(\frac{1+e^{a+b}}{e^a+e^b}\right)$.
It's a fun exercise to check the following properties:
- $a\boxplus b=b \boxplus a$
- $a\boxplus b=0$ iff $a=0$ or $b=0$
- $a \boxplus (-b) = (-a) \boxplus b= -(a \boxplus b)$
- $a \boxplus \infty = a$, in the sense that $\displaystyle \lim_{x \to \infty} a\boxplus x=a$
- $(a\boxplus b) \boxplus c=a \boxplus (b \boxplus c)$
So just looking at these properties I though maybe $(\mathbb{R},+,\boxplus)$ is a commutative ring without identity. But it doesn't satisfy the distributive law. Is there anything that can be said about such a structure?
Edit: After a helpful comment from Bill I would like to point out which of these properties I find 'ringlike'. Property 3 is a statement that is true in a ring and doesn't make sense in just the semigroup $(\mathbb{R},\boxplus)$. Property 2 is the definition of an integral domain (if it was a ring). So it seems to me that having $-$ and $0$ puts me in the mind of connecting $\boxplus$ with $+$. But I don't know if this is necessary: is there some theory of semigroups with an extra unary negation operator? Maybe that's all I need.