On a quasi-ordering on Boolean functions

It was proved few years ago that classes of Boolean functions definable by means of functional equations [9], or equivalently, by means of relational constraints [15], coincide with initial segments of the quasi-ordered set (Ω,≤) made of the set Ω of Boolean functions, suitably quasi-ordered. The resulting ordered set (Ω/ ≡,⊑) embeds into ([ω] ,⊆), the set -ordered by inclusionof finite subsets of the set ω of integers. We prove that (Ω/ ≡,⊑) also embeds ([ω],⊆). We prove that initial segments of (Ω,≤) which are definable by finitely many obstructions coincide with classes defined by finitely many equations. This gives, in particular, that the classes of Boolean functions with a bounded number of essential variables are finitely definable. As an example, we provide a concrete characterization of the subclasses made of linear functions.