On the number of clonoids

TL;DR

This paper investigates the number of clonoids, a generalization of clones, showing that there are continuum many on a 2-element set, contrasting with the countably many clones, and extends the analysis to finite sets and 2-element algebras.

Contribution

It establishes the cardinality of clonoids for finite sets and 2-element algebras, revealing a rich structure beyond classical clone counts.

Findings

Countably many clones on a 2-element set.

Continuum many clonoids on a 2-element set.

Cardinality results for clonoids from finite sets to 2-element algebras.

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 $\mathbf{B}$.