Problem 20
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Transpose symmetry for injectivity over semirings
Let \(n\geq 3\). Let \(R\) be a commutative semiring with \(0\) and \(1\), and let \(A\in M_n(R)\). Denote by
\[ \lambda_A:R^n\longrightarrow R^n,\qquad v\longmapsto Av \]
the \(R\)-linear endomorphism defined by left multiplication by \(A\).
Open problem
Is it true that
\[ \lambda_A \text{ is injective} \quad\Longleftrightarrow\quad \lambda_{A^{\mathrm t}} \text{ is injective} \]
for every commutative semiring \(R\) and every \(A\in M_n(R)\)?
The case \(n=2\) has a positive answer. The first open case is therefore \(n=3\).
Verification notes. MathOverflow question: Transpose symmetry of injectivity of linear maps over semirings. Lean statements: Lean live code.
Source: Junyan Xu.
LaTeX source
\begin{problem}[Transpose symmetry for injectivity over semirings]
Let \(n\geq 3\). Let \(R\) be a commutative semiring with \(0\) and \(1\), and
let \(A\in M_n(R)\). Denote by
\[
\lambda_A:R^n\longrightarrow R^n,\qquad v\longmapsto Av
\]
the \(R\)-linear endomorphism defined by left multiplication by \(A\).
Is it true that
\[
\lambda_A \text{ is injective}
\quad\Longleftrightarrow\quad
\lambda_{A^{\mathrm t}} \text{ is injective}
\]
for every commutative semiring \(R\) and every \(A\in M_n(R)\)?
The case \(n=2\) has a positive answer. The first open case is therefore
\(n=3\).
\end{problem}
\paragraph{Verification notes.}
MathOverflow question:
https://mathoverflow.net/questions/511862/transpose-symmetry-of-injectivity-of-linear-maps-over-semirings
Lean statements:
https://live.lean-lang.org/#codez=JYWwDg9gTgLgBAWQIYwBYBtgCMBQO0Cm0BIcBAHsAM4xUD6mAdgUlAJKMAmBYBXfMOlAgB3OowiCAxhHRwAXDjjK4gYCI4ACkZwASgrgAVAJ68AlJrr6AwhBAgAyiWBRgjAOa7zGgIL7kMF3I4bW0dUwAaJRUAGVcWdn5efkZ4PW8AOhk5QHIiOAAauFjmVg5uJO4U3TgM4REFAF44LCMoqmgoFvxUIigSMkoaeiZ40p4%2BCsFa8Uk6LIUo5XUtKvlDEwIvS1WbO0cQZ1cPMM1fVf9AzQAxVzgAZi9r7XvPSJVCuJLE8YEqjLncgpFEZfZKparpWoNJotZRtKAdIA
Added July 4, 2026.