Clones of Borel Boolean Functions

TL;DR

This paper explores the structure of Borel clones on the Boolean domain, extending Post's classification to countable arities and revealing complex relationships between finitary and Borel clones.

Contribution

It extends Post's classification of Boolean clones to Borel clones with countable arities and analyzes their lattice structure and extensions.

Findings

Finitary affine clones have unique Borel extensions.

Certain clones contain finitely many Borel clones.

The full structure of Borel clones may be highly complex.

Abstract

We study the lattice of all Borel clones on $2 = \{0,1\}$: classes of Borel functions $f : 2^n \to 2$, $n \le \omega$, which are closed under composition and include all projections. This is a natural extension to countable arities of Post's 1941 classification of all clones of finitary Boolean functions. Every Borel clone restricts to a finitary clone, yielding a "projection" from the lattice of all Borel clones to Post's lattice. It is well-known that each finitary clone of affine mod 2 functions admits a unique extension to a Borel clone. We show that over each finitary clone containing either both $\wedge, \vee$, or the 2-out-of-3 median operation, there lie at least 2 but only finitely many Borel clones. Over the remaining clones in Post's lattice, we give only a partial classification of the Borel extensions, and present some evidence that the full structure may be quite complicated.