Equational definability and a quasi-ordering on Boolean functions

Earlier work by several authors has focused on defining Boolean function classes by means of functional equations. In [10], it was shown that the classes of Boolean functions definable by functional equations coincide with initial segments of the quasi-ordered set (Ω,≤) made of the set Ω of Boolean functions, suitably quasi-ordered. Furthermore, the classes defined by finitely many equations coincide with the initial segments of (Ω,≤) which are definable by finitely many obstructions. The resulting ordered set (Ω/ ≡,v) embeds into ([ω]<ω ,⊆), the set -ordered by inclusionof finite subsets of the set ω of integers. But the converse also holds. We define an order-embedding of ([ω]<ω,⊆) into (Ω/ ≡,v ). From this result, we deduce that the dual space of the distributive lattice made of finitely definable classes is uncountable. Looking at examples of finitely definable classes, we consider classes of Boolean functions with a bounded number of essential variables and classes of functions with bounded polynomial degree. We provide concrete equational characterizations of these classes, as well as of the subclasses made of linear functions with a bounded number of essential variables. Moreover, we present descriptions of the classes of functions with bounded polynomial degree in terms of minimal obstructions.