7 problems

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 $E_k$ 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

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.