On the lattice of equational classes of Boolean functions and its closed intervals

Abstract: Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boolean case |A|=2, the lattice E_A contains uncountably many equational classes, but only countably many of them are clones. The aim of this paper is to provide a better understanding of this uncountable lattice of equational classes of Boolean functions, by analyzing its "closed" intervals" [C_1,C_2], for clones C_1 and C_2. For |A|=2, we give a complete classification of all such closed intervals in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of E_A.