Clones of Borel Boolean Functions

Abstract

We study the lattice of all Borel clones on $2=\{0,1\}$: classes of Borel functions $f:2^n\to 2$, $n\le \ddot{I}‰$, 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.

58 pages, 14 figures