Small clones and the projection property

TL;DR

This paper extends the classification of minimal clones from finite to infinite universes and multiclones, revealing structural properties and connections to Boolean groups.

Contribution

It generalizes the classification of minimal clones to infinite universes and multiclones, and links clone structures to Boolean groups.

Findings

Every non-trivial clone contains a small clone of one of five types.

If a clone contains all constant operations, then binary idempotent operations are projections.

Existence of non-projection idempotent operations relates to Boolean group structures.

Abstract

In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a "small" clone of one of the five types. From it we deduce, in part, an earlier result, namely that if $\mathcal C$ is a clone on a universe $A$ with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some $m$-ary idempotent operation is not a projection some $m\geq 3$ if and only if there is a Boolean group $G$ on $A$ for which $\mathcal C$ is the set of all operations $f(x_1,..., x_n)$ of the form $a+\sum_{i\in I}x_i$ for $a\in A$ and $I\subseteq \{1,..., n\}$.