On the number of clonoids

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 $A = B = {0,1}$\mathrm{.}Moregenerally,foranyfiniteset$A$andany$2$-elementalgebra$$\mathbf{B}$$,wegivethecardinalityofthesetofclonoidsfrom$A$to$$\mathbf{B}$$thatareclosedundertheoperationsof$$\mathbf{B}$$.