By a rig, I mean a ring whose elements don't necessarily have additive inverses, sometimes called a semiring. I want to ask very broad and potentially very naive question about solving quadratic equations in rig theory.
Suppose we're given:
- a commutative rig, call it $R$
- a pair of elements of $R$, call them $a$ and $b.$
Now define: $$X = \{x \in R \mid x^2 = ax+b\}$$
Questions.
Q0. Are there any theorems around that tell us anything interesting about $X$? e.g. which put bounds on its cardinality or tell us something interesting about how $X$ "sits inside" $R$?
Q1. Are there any techniques available for finding the elements of $X$ explicitly?
Q2. If the answers are "no", are there constraints we can put on $R$ that don't imply that addition in $R$ is cancellative, such that the answers to the above questions become "yes"?
I'm also interested in the potentially harder problem of finding $$\{x \in R \mid x^2+a'x+b' = ax+b\}$$
for $a',b',a,b \in R$.
