16 problems

Bi-UF Positive Conjecture. The prototypical semiring $\mathbb{N}_0$ is the only positive semiring that is a bi-UFS.

Cancellative-elements conjecture. An element of $S[\mathcal{M}]/\operatorname{Bend}(J)$ is cancellative if and only if it is the image of a term of $S[\mathcal{M}]$.

Let $[n]=\{1,\ldots,n\}$, let $\mathfrak{S}_n$ be the set of permutations of $[n]$, and let $\mathfrak{C}_n$ be the set of $n$-cycles in $\mathfrak{S}_n$. A join/union expression i…

Geometricity conjecture. Any semiring homomorphism $\nu:R\to\mathbb{Z}^B_{\mathrm{pos}}\cup\{-\infty\}$ is geometric.

Geometricity conjecture. Every morphism $\nu:\varphi_Y\to\varphi_X$ is geometric.

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 $S$ be a congruence-simple semiring with a multiplicatively absorbing element $w$, and suppose that … for every $x\in S\setminus\{w\}$. Classification conjecture. If $S$ is fin…

Omega-primality finiteness conjecture. One has

Generalised regularity classification. The displayed classification summarises the generalised regularity properties of $M_n(\mathfrak{L})$, $UT_n(\mathfrak{L})$, and…

Let $G$ be a finite group, let $\mathbb{N}[G]_n$ denote the $n$-th grade of the group semiring, and let $\operatorname{Comp}(\mathbb{N}[G])_n$ denote its composite elements in that…

Let $A=\mathbb{Z}[T_1,\dots,T_n]$, and let $A^+=\mathbb{N}[T_1,\dots,T_n]$. Let $B$ be a subring of $A$ and $I$ an ideal of $B$. Write $QF(B)$ for the quotient field of $B$. Kala's…

A semifield is a commutative semiring whose multiplicative structure is a group. A semiring is finitely generated as a semiring when it is generated by finitely many elements under…

Idempotency conjecture. Every parasemifield which is finitely generated as a semiring is additively idempotent.

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…