12 problems

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

Let $A$ be a set and let $\rho$ be a partial order on $A$ with least element $0$ and greatest element $1$. Let $\operatorname{Pol}\rho$ be the clone of operations preserving $\rho$…

Let $A$ be a set and let $\rho$ be a partial order on $A$ with least element $0$ and greatest element $1$. Set $F=\operatorname{Pol}\rho$, the maximal clone of operations preservin…

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…

The three-submaximal-clones conjecture. The poset $\mathfrak P_3$ has exactly three submaximal elements, namely the minor-equivalence classes of $\mathcal C_2$, $\mathcal C_3$, and…

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

A clone is a set of operations containing projections and closed under composition. A clone is finitely related if it is the polymorphism clone of a finite set of relations, and fi…

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…

Let $A$ be a finite set with $\lvert A\rvert=k\geq 3$, and let $F$ be a centraliser clone on $A$, meaning that $F^{**}=F$. The $n$-ary part of $F$ is denoted by $F^{(n)}$, and $F$…

Let $I_{\bar d=0}$ be the ideal of subsets of $\mathbb N$ having upper density zero, and let $\mathscr C_{I_{\bar d=0}}$ be the clone of all finitary functions on $\mathbb N$ that…