80 problems

Let $G_n(\mathbb{C})$ denote the set of pairs $(A,B)$ of $n\times n$ complex matrices that generate the matrix ring, and let $\operatorname{msl}_{\mathbb{C}}(A,B)$ be their matrix…

Let $1\leqslant p<\infty$, let $X_p$ be the underlying $L^p$ phase-space space, and let $L^p_+$ and $L^p_-$ denote the outgoing and incoming boundary spaces. Let…

Correspondence conjecture. The Green's relations on a subsemigroup $U$ of $S$ have as counterparts prolonged images in $S$, namely the fine equivalence relations on $S$.

Krohn–Rhodes-like decomposition conjecture. Every propositional Krom logic program can be sequentially decomposed into aperiodic and permutation programs.

Restriction conjecture. Every automorphism of $\mathcal P(H)$ maps $\mathcal P_1(H)$ bijectively onto itself and therefore restricts to an automorphism of $\mathcal P_1(H)$.

Inner automorphism conjecture. The automorphisms of $\mathcal P(\mathbb Z)$ are all inner, and the automorphisms of $\mathcal P_0(\mathbb Z)$ are all inner.

Bender et al.'s conjecture. If $G(S)$ is uniquely complemented with clique number $3$ or greater and every vertex has a unique complement, then $G(S)$ is isomorphic to…

Tamura's conjecture. The implication holds for arbitrary semigroups $S_1$ and $S_2$.

Pseudo $\ll$-transitivity conjecture. If

Let $S$ be a numerical semigroup, and call a nonzero small element of $S$ lonely if it has the property specified in the source. Lonely element conjecture. If every nonzero small e…

Let $M$ be a finite monoid, and let $K_{\rm G}(M)$ denote its $\rm G$-kernel, the intersection of $\mu^{-1}(1)$ over all relational morphisms from $M$ to finite groups, where…

Chueshov's conjecture. The semigroup $\{e^{\mathcal{A}t}\}_{\mathcal{L}(\mathcal{H})}$ is analytic in an appropriate sector of the complex plane.

Right-ideal characterization conjecture. If $1_{\mathbf{P_{PU}}}$ belongs to $\mathbf{A}$, then

Let $S=(S,\cdot)$ and $R=(R,\cdot)$ be semigroups, with $R$ admitting a left invariant mean, and let $A\subseteq R$ be piecewise syndetic. For commuting homomorphisms…

Torsion–almost-divisibility conjecture. The semigroup $S(+)$ is torsion if and only if $S(+)$ is almost-divisible.

Countable subdirect product conjecture. The answer to the first question is affirmative, while the answer to the second question is negative: there exist infinite non-periodic semi…

Least weakly separative congruence conjecture. The relation $\sigma_S$ is the least weakly separative congruence.

Let $M_4 = \operatorname{Mon}\langle a,b\mid aba^2=ba\rangle$ and let $M_7$ denote its anti-isomorphic one-relation monoid. A faithful tropical representation of a monoid is an inj…

Cancellative free-product conjecture. The semigroup $S$ is a free product of its amenable subsemigroups.

Finite-index free-product conjecture. The semigroup $S$ contains a finite-index subsemigroup which is a free product of its right amenable subsemigroups.

Sørensen's conjecture. Every right cancellative left amenable semigroup is left cancellative.

Conjecture. Under these assumptions, $S$ satisfies the quasi-identities $(,)$.

Let an SBC (set-based category) be a category with subobjects equipped with a functor to $\mathbf{Set}$ satisfying the stated embedding, image-factorisation, and common-subobject c…

Let $S$ and $T$ be finite semigroups, and let $S\star T$ denote their free product. An automaton semigroup is a semigroup generated by the states of a finite synchronous automaton…

Let $S$ be a finitely generated semigroup of rational maps, and say that a rank $2$ free subsemigroup is a free subsemigroup generated by two elements. An invariant probability mea…