25 problems

Rank-generating realization conjecture. There exists a finitely generated LCH monoid $M$ such that

Multiplicability conjecture for finitary upho lattices. Every finitary upho lattice, and in particular every finite-type $\mathbb{N}$-graded upho lattice, is multiplicable.

Fu–Peng–Zhang's multiplicability conjecture. Every finitary upho poset, in particular every finite-type $\mathbb{N}$-graded upho poset, is multiplicable.

Fu–Peng–Zhang's coloring conjecture. Every finitary upho poset, in particular every finite-type $\mathbb{N}$-graded upho poset, has a coloring.

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

Let $T\mathcal{P}_n$ be the tied party monoid, and let $\mathcal{EP}_n(p,q)$ be the tied Party-Hecke algebra generated by $J_1,\ldots,J_{n-1}$, $F_1,\ldots,F_{n-1}$, and…

Monoid realization conjecture. The monoid $M$ is left-cancellative, every pair of elements of $M$ has a greatest common left divisor, and therefore…

Finite catenary-degree conjecture. If $F_{\mathrm{red}}$ is finitely generated, then

Greedy 0-monoid conjecture. The formal power series $g(x)$ is a regular semi-upho function if and only if it admits an infinite greedy $0$-monoid series $\{M_k^0\}_{k\ge 1}$.

Regularity conjecture. Every finitary semi-upho poset is regular.

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

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{F}$ be the monoid of functions under composition, and let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a…

Let $\mathbf{M}$ denote the class of metric spaces, let $\mathbf{P}_{\mathbf{X}}$ be the monoid associated with a metric space $\mathbf{X}$, and let $\mathbf{Am}$ be the monoid of…

Equivariant monoid realization conjecture. Any $G$-connected based $G$-space $X$ is weakly equivalent to

Let $G(n,k)$ be a generalized Petersen graph, and let ${\rm Cay}(M,C)$ denote a monoid Cayley graph with connection set $C$ of size $2$. The monoid Cayley characterization conjectu…

Let $n\geq 3$. The monoid $\mathcal{H}_n^+$ is defined by the presentation … Let $\mathcal{B}_{n+1}$ be the braid group on $n+1$ strands, with standard generators…

Let $G$ be a finite abelian group, and let $H$ be a transfer Krull monoid over $G$, meaning that there is a transfer homomorphism $H\to\mathcal B(G)$ to the monoid of zero-sum sequ…

Let $M$ be a matroid of rank $k$ on $n$ elements, and let $d4c9_p(M)$ denote its associated monoid. Write $\operatorname{Irr}(\mathcal{Q}_p(M))$ for the set of irreducibles of this…

Cancellativity conjecture. The monoid $S(X,r)$ is cancellative if and only if $r$ is involutive.

Closed-upper-set characterization. An upper set $Z$ of $\Lambda^{\ast}$ is closed if and only if it satisfies the four rules.

Let $S$ be a finitely presented monoid of linear growth. A free sandwich is a subset of the form $p\langle q\rangle r$, where $p,q,r$ are words and $\langle q\rangle$ denotes the s…

Injectivity conjecture. The natural map

Let $\Lambda^*$ be the cross section lattice of a $J$-irreducible monoid $M$, and let $J_0$ and $n$ have the meanings established in the preceding theorem. Characteristic-polynomia…

Let $\Lambda$ be a distributive cross section lattice of the $J$-irreducible monoid of the pair $(G_0,\rho)$, where $G_0$ is a simple algebraic group of one of the types $A_{n-1}$,…

Presentation conjecture. Proposition $$ gives a presentation by generators and relations of the monoid $\overline{\square}([n],[n])$ for every $n\geqslant 2$.