Miguel Couceiro

Let A be a finite set with | A |&Mac179;2. The composition of two classes I and J of operations on A, is defined as the set of all composites f (g₁ ...,g_n ) with f ŒI and g₁ ...,g_n ŒJ . 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 (2^¿₀ ) 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 idempotent classes C ₁ and C ₂.For | A |=2, we give a complete classification of all closed intervals [C ₁, C ₂] in terms of their size, and provide a simple, necessary and sufficient condition characterizing the uncountable closed intervals of E_A.