115 problems

A Robbins algebra is an algebra satisfying the Robbins identities. Robbins algebra conjecture. Every Robbins algebra is a Boolean algebra. McCune proved this conjecture using the E…

Park's conjecture. Every finitely generated variety of finite type that has a finite residual bound is finitely based.

Countability conjecture. The minor equivalence relation on clones on finite domains has only countably many classes.

Let $\bA$ be an algebra with a Taylor term in its clone, and let $R$ be a finite set of relations compatible with $\bA$. Write $CSP(R)$ for the constraint satisfaction problem over…

Let $\mathbb{A}=\langle A;\mathscr{R}\rangle$ be a finite core relational structure of finite relational signature. A polymorphism of $\mathbb{A}$ is an operation that is a homomor…

A reduction from general finite constraint satisfaction problems to digraph constraint satisfaction problems maps a relational structure to a digraph while preserving the relevant…

Let $\mathbb{A}$ be a finite relational structure that is a core, meaning that it has no proper retract. Its algebra of polymorphisms consists of the operations on its domain prese…

Complex-subalgebra characterization conjecture.

Entropicity conjecture. Every idempotent algebra $(A,f)$ with the generalized entropic property is entropic.

Complex-subalgebra variety conjecture. The variety generated by these complex algebras coincides with ${\mathcal V}$ if and only if ${\mathcal V}$ has a basis of linear and idempot…

Conjecture for idempotent groupoids. The generalized entropic property and the entropic law are equivalent for idempotent groupoids.

Similarity–equivalence conjecture. $H_1$ and $H_2$ are similar if and only if they are equivalent.

Flexibility-free approximate polymorphism conjecture. The Alekseev--Filmus theorem should hold without the assumption that $P$ is flexible, for any alphabet $\Sigma$: for every suf…

Let $\mathbf{a}$ be a finite idempotent algebra, and let $\Gamma(\mathbf{a})$ be its associated relational clone. The source has established that if $\mathbf{a}$ has no Taylor term…

Let $\mathbf{a}$ be a finite idempotent algebra, and let $\Gamma(\mathbf{a})$ denote the constraint language consisting of the relations compatible with all operations of…

Bounded-arity symmetric-part conjecture. For any lattice variety axiomatized by identities of arity at most $k$, one has

Symmetric-part conjecture. Each lattice variety $\mathcal{V}$ equals its symmetric part:

Let $cmathcal{V}$ be a symmetric strongly irregular variety. Its balanced regularisation is the variety obtained by imposing balanced identities in the corresponding regularisation…

Finiteness conjecture. There are only finitely many clonoids from $\mathbf{A}$ to $\mathbf{B}$ if and only if $\mathbf{A}$ and $\mathbf{B}$ are of coprime order.

Automorphic–geometric equivalence conjecture. In every subvariety $\Theta$ of the variety of all groups, automorphic equivalence coincides with geometric equivalence.

Automorphic–geometric equivalence conjecture. Despite the fact that

For every cardinal $\kappa$, let $\mathsf{A}_\kappa$ be the Heyting algebra whose lattice reduct is obtained by adding a new maximum $1$ to the powerset lattice…

Let $\mathbb{SH}$ denote the variety of connexive semi-Heyting algebras, let $\mathbf A\in\mathbb{SH}$ satisfy (AT1), and let $x,y\in A$. The conjecture. The identity … holds in…

Let $\mathbb A$ be a first-order reduct of a finitely bounded homogeneous structure. The constraint satisfaction problem of $\mathbb A$ is denoted by…

Dibinary extension-monad conjecture. The extension monad of $cmathbb{A}$ has the same underlying set as the extension monad of its $cbeta$ reduct.