Let $S$ be the semiring in $\mathbb{R}[x,y]$ generated by the nonnegative real numbers $\mathbb{R}\_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$, i.e.
$$S = \left\{ \sum\_{i,j,k,\ell \ge 0 } a\_{i,j,k,\ell} x^{i} y^{j} (x + y - 1)^{k} (1 - (x+y))^{\ell} : \, a\_{i,j,k,\ell} \ge 0 \,\, \forall i,j,k,\ell \,\right\}.$$
Let $\pi$ be the quotient map $\mathbb{R}[x,y] \twoheadrightarrow \mathbb{R}[x,y]/(x + y - 1)$.


Clearly $S \subseteq \pi^{-1}(\pi(S))$. Is the inclusion strict?



The semiring generated by $\mathbb{R}_+$ and the polynomials $x, y, x + y - 1, 1 - (x+y)$
The question can be reformulated as: *Does $\pi^{-1}(0)$ lie in $S$?* Indeed, if so, then for all $s\in S$, $t\in \mathbb R[x,y]$ with $\pi(s)=\pi(t)$ we have $t-s\in S$, so $t=s+(t-s)\in S$ as well.


And this seems to be true. Actually, every $t\in \pi^{-1}(0)$ is a linear combination of the polynomials $m\_{ij}=x^iy^j(x+y-1)$; but both $m\_{ij}$ and $-m\_{ij}$ lie in $S$, thus $t\in S$ as well.


