2
$\begingroup$

Consider the Tropical semiring $\mathbb{T} = (\mathbb{R}\cup\{-\infty\},\min,+)$.

The Tropical semiring is also a local semiring? In other words, $\mathbb{T}$ has a unique maximal ideal?

$\endgroup$
2
$\begingroup$

Let $I$ be an ideal of $\mathbb{T}$. If $I$ contains a real $r$, then it contains every real $s$ since $s = r + (s-r)$. Moreover, it contains $-\infty$ since $r + (-\infty) = -\infty$. Thus $I = \mathbb{T}$. It follows that $I = \{-\infty\}$ is the unique proper ideal of $\mathbb{T}$. Indeed, it is closed under $\min$ and, for every $r \in \mathbb{T}$, $r + (-\infty) = -\infty$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.