Generation of the Post Lattice by irreducible clones

Abstract

Clones of logic functions form an algebraic lattice, which in the case of Boolean functions was completely described by E.L.Post in 1941. This lattice, often referred to as the Post Lattice, has been well studied from various angles, particularly, the generation of the Post Lattice by its subsets. This paper discusses the results about clones that are irreducible by means of meet- and/or join-operations of the lattice. We show that the join-irreducible clones generate the Post Lattice, and dually, the meet-irreducible clones generate the Post Lattice. We present a complete description of such generation in both cases. We observe that any clone can be presented as a join or meet of at most two irreducible clones. In the former case each clone is the join of at most four join-irreducible clones and in the latter case each clone is the meet of at most three meet-irreducible ones.