36 problems

Bi-UF Positive Conjecture. The only such positive semiring is the prototypical semidomain $\mathbb{N}_0$.

Converse conjecture. The converse of Proposition 3.14 is false: not every semi-inner automorphism of $(_{R}\mathcal{M}^{0})$ is equinumerous.

Existence conjecture. There exists such an $R$ having a semi-inner, non-equinumerous automorphism $\varphi\in\operatorname{Aut}(_{R}\mathcal{M}^{0})$.

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 $S[\mathcal{M}]$.

Hypothesis R. Every null root of every polynomial over $(\mathcal A,\mathcal A_0)$ is a factor-root.

Let $n\geq 1$, let ${\mathbb{B}}[x_1,\ldots,x_n]$ be the polynomial semiring over the Boolean semifield, and write $X^{\vec{a}}=x_1^{a_1}\cdots x_n^{a_n}$ for…

Alarcón–Anderson's classification conjecture. If $\mathcal{P}$ is a prime ideal in ${\mathbb{B}}[x]$ not containing $\langle x\rangle$, then…

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…

Let $S_{\rm res}^d$ be a semiring of matrices with support restrictions, and let $U$ be a matrix whose columns all sum to one. A matrix $U$ is power universal if it satisfies the r…

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.

Fullness conjecture. The functor $\Phi$ is full; equivalently, every morphism $u:varphiYtovarphiX$ is geometric and is induced by a morphism $mu:Xto Y$ of tropical fans.

Geometricity conjecture. Any semiring homomorphism between the Boolean function semirings on 1-dimensional tropical fans is geometric.

Let $S$ be a semiring and let $M$ be an $S$-semimodule. A right congruence on $S$ is regular if it is compatible with the right action of $S$, and it is maximal regular if it is ma…

Conjecture. The three conditions are equivalent for every additively cancellative semiring $S$.

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…

Let $S$ and $S'$ be semirings with at least two elements and with bi-absorbing elements $o$ and $o'$, respectively. For the box-product $S\boxtimes S'$, write $V(G)$ for the semiri…

Let $A$ and $B$ be finite sets. Let $R\subset \mathbb Z^A_{\mathrm{pos}}\cup\{-\infty\}$ and $S\subset \mathbb Z^B_{\mathrm{pos}}\cup\{-\infty\}$ be subsemirings. A homomorphism…

Omega-primality finiteness conjecture. One has

Chapman–Gotti–Gotti conjecture. For every non-integer rational $\alpha>1$, the set of elasticities of $\mathbb{N}_0[\alpha]$ is

Let $t\in \mathbb{N}$ be arbitrary. Write $K^{\infty}_t$ for the finite min-plus semiring with entries drawn from $\{0,\ldots,t,\infty\}$, and let $M_3(K^{\infty}_t)$ denote the mo…

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

Il'in's conjecture. Every left (right) $S$-semimodule has an injective hull if and only if $S$ is additively regular. This conjecture characterizes semirings for which the category…