47 problems

Complex-subalgebra characterization conjecture.

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

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:

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

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

Extension-monad conjecture. The extension monad of a ring has the same underlying set as the extension monad of its additive reduct.

Let $H$ be a connected core on $k\geq 3$ vertices. Write $E_H$ for the edge relation of $H$, let $\langle E_H\rangle$ denote the relational clone generated by $E_H$, and let…

Solvability and abelianness conjectures. (i) Every finite superconnected quandle is solvable. (ii) Every finite simple superconnected quandle is abelian.

Let $\Gamma$ be a constraint language. It has short pp-definitions if every $n$-ary relation pp-definable from $\Gamma$ is definable by a primitive positive formula whose length is…

Let $E_k$ be a $k$-element set with $k\geq 2$, and let $\mathcal S$ be a clone over $E_k$. Say that $\mathcal S$ satisfies $\operatorname{TS}(n)$ for every $n\geq 2$ and satisfies…

Three strongly connected components conjecture. The digraph $D_{\mathbb{A}}$ has at least three strongly connected components.

Three-element generated subalgebra conjecture. Under these hypotheses,

Two-element projection quotient conjecture. For every $a\ne b\in\mathbb{A}$, there is a surjective homomorphism

Let $\mathbb{A}$ be a minimal Taylor algebra, meaning a Taylor algebra whose clone is minimal in the sense used by the source, and suppose that $\mathbb{A}$ is generated by two ele…

Let $\mathbb{A}$ be a finite idempotent algebra, meaning that every basic operation satisfies $f(x,\ldots,x)=x$. Bounded relational-width term conjecture. The algebra $\mathbb{A}$…

Let $\mathbb{A}$ be a finite algebra, and say that it has few subpowers when the equivalent conditions in the source's Few Subpowers theorem hold. Given a finite subset…

Let $\Gamma$ be a constraint language, and let $r(n)$ be the logarithm of the number of distinct $n$-variable relations definable by primitive positive formulas over $\Gamma$. Chen's…

Let $F$ be freely generated by $\\{x_i:i<\omega+\omega\\}$, and let $\langle x_i:\omega\leq i<\omega+\omega\rangle_F$ denote the subalgebra generated by those elements. Shelah's au…

Let $\mathbb{H}$ be a finite structure with finite relational signature, and let its polymorphisms be the operations preserving the relations of $\mathbb{H}$. A Noname chain is the…

Let $\mathbb{H}$ be a finite structure with finite relational signature, and let its polymorphisms be the edge-preserving operations on its domain. A Kearnes–Kiss chain is the chai…

Odd-antlattice quasivariety conjecture. The quasivariety of odd antilattices is proper; that is, it is not a variety.

Three-element reduction conjecture. Every clone on three elements that does not admit a minor-preserving map from ${\mathbf I}$ has a minor-preserving map to one of the following t…

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