On the number of clonoids (Q2009230)

A clonoid $C_{A,B}$ is a set of finitary functions from a set $A$ to a set $B$ that is closed under taking minors. The author studies the size of clonoids for finite sets and algebras. For any finite set $A$ and any two-element algebra $B$, $C_{A,B}$ is finite iff $B$ has an NU-term, it is countably infinite iff $B$ has a Mal'cev term but no majority term and it has size continuum otherwise. If $A$ is a finite set and $B$ a finite idempotent algebra then $C_{A,B}$ has size continuum iff $B$ has no cube term. If $B$ has a cube term, then there are countably many such clonoids.