On Composition-Closed Classes of Boolean Functions

We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification (diagonalization) of variables and under introduction of inessential variables (cylindrification), but they do not necessarily contain projections. Thus the lattice formed by these classes is an extension of the Post lattice. The cardinality of this lattice is continuum, yet it is possible to describe its structure to some extent.

I. Introduction and Preliminaries

The goal of this paper is to describe classes of Boolean functions that are closed under composition, diagonalization, cylindrification and permutation of variables, but do not necessarily contain projections, thereby generalizing Post's description of Boolean clones. In order to formulate our problem more precisely, we first recall some definitions and introduce some notation. For more background on clone theory, we refer the reader to the monographs [1] and [2].