44 problems

An asynchronous transition system is a concurrent-process model equipped with events and an independence relation; its homological dimension is the largest degree in which its asso…

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…

Bi-${\rm FP}_\infty$ conjecture. Every one-relator monoid of the form $\langle A\mid u=v\rangle$ is of type bi-${\rm FP}_\infty$.

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

Let a Schreier split extension of monoids be a Schreier extension whose underlying short exact sequence is split, and let a split epimorphism of monoids be an epimorphism admitting…

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 $\{M_k^0\}_{k\ge 1}$.

Greedy LCH-series conjecture. The propositions establishing that an infinite greedy LCH monoid series produces a regular upho function, and that the associated coefficient inequali…

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

Regularity conjecture. Every finitary upho poset is regular.

Algebraic Kirchberg–Phillips conjecture. If there is a $\mathbb{Z}$-isomorphism $\phi:T_E\to T_F$ such that

Lunar–SAP equivalence conjecture. The local algebraic lunar condition on $\mathcal{M}$ is equivalent to the global analytic SAP of its associated Hankel system…

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

A delta lens is a delta lens $(f,\varphi)\colon A\to B$, and a split opfibration is a split opfibration between categories. For monoids, viewed as categories with a single object,…

Reduced-twin conjecture. There exists no reduced twin of non-cancellative tuples when the corresponding second homotopy group vanishes.

Higher monoid cohomology vanishing conjecture. The same method should yield

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…