4 problems

Multiplicative idempotence conjecture. If $S$ is multiplicatively divisible, then $S$ is multiplicatively idempotent.

Classification conjecture. Every multiplicatively idempotent congruence-simple semiring is finite and isomorphic to one of the semirings $\mathbb{S}_1$, $\dots$, $\mathbb{S}_8$.

Let $(R,\vee,\circ)$ be a finite simple additively idempotent semiring with an absorbing greatest element, and suppose it possesses a finite idempotent irreducible semimodule witho…

Let $(R,+,\cdot)$ be a finite, simple, additively idempotent semiring, and let $(M,+)$ be a finite idempotent $R$-semimodule. A semimodule is sub-irreducible if it is non-quasitriv…