39 problems

Let $X$ be a set, and let $\mathcal{T}(X)$ denote the full transformation semigroup on $X$. For the notation $N^{X}_{x_1,\ldots,x_t}$ and the functions $\xi$ and $\alpha$, use the…

Let $S$ be the finite semigroup under consideration, let $u,u'\in S$, let $u^{{+}_r}$ and ${u'}^{{+}_l}$ denote their associated right and left idempotents, let $\boldsymbol{*}$ de…

Let $\mathcal{J}_n$ be the Jones monoid, let $E_n$ be its set of idempotents, let $\operatorname{rank}(\alpha)$ denote the rank of $\alpha\in E_n$, let $e_n=|E_n|$, and let $d_n^*$…

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…

Tringali–Yan conjecture. If $S$ is a numerical monoid properly contained in $\mathbb{N}$, then the automorphism group of $\mathcal{P}_{0}(S)$ is the trivial group:

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…

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.

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.

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…

Let $S$ be a semigroup of rational maps. An invariant probability measure is a probability measure invariant under every element of $S$, and a non-injective element is a map in $S$…

Trivial-range conjecture. The restriction operator $R$ has trivial range. The conjecture concerns the restriction of automorphisms from the superextension of $\mathrm{M}_{r,m}$ to…

Let $\mathrm{M}_{r,m}$ denote a finite monogenic semigroup, let $\lambda(S)$ denote its superextension, and let $\operatorname{Aut}(G)$ denote the automorphism group of a group $G$…

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

Let ${\mathcal O}_n$ be the origami monoid of degree $n$ defined by the DNA-origami rewriting system. Finiteness conjecture. The order of ${\mathcal O}_n$ is finite for all $n$. Th…

Conjugacy-class cardinality conjecture. The conjugacy class of $e$ has

Let $W(\ell_a,\ell_a)$ be the set of binary words containing $\ell_a$ occurrences of each letter, and consider its $\mathcal{UT}_2(\mathbb{T})$ equivalence classes. Generalised Cat…