  We consider classes of Boolean functions stable under compositions both from
the right and from the left with clones. Motivated by the question how many
properties of Boolean functions can be defined by means of linear equations, we
focus on stability under compositions with the clone of linear idempotent
functions. It follows from a result by Sparks that there are countably many
such linearly definable classes of Boolean functions. In this paper, we refine
this result by completely describing these classes. This work is tightly
related with the theory of function minors, stable classes, clonoids, and
hereditary classes, topics that have been widely investigated in recent years
by several authors including Maurice Pouzet and his coauthors.
