On the number of clonoids

Abstract

A clonoid is a set of finitary functions from a set $A$ to a set $B$ that is closed under taking minors. Hence clonoids are generalizations of clones. By a classical result of Post, there are only countably many clones on a 2-element set. In contrast to that, we present continuum many clonoids for $A = B = \{0,1\}$. More generally, for any finite set $A$ and any $2$-element algebra $\mathbf{B}$, we give the cardinality of the set of clonoids from $A$ to $\mathbf{B}$ that are closed under the operations of $\mathbf{B}$.