Completely distributive CSL algebras with no complements in C-p

Anoussis and Katsoulis have obtained a criterion for the space Alg L boolean AND C-p to have a closed complement in C-p, where L is a completely distributive commutative subspace lattice. They show that, for a given L, the set of p for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice C for which Alg L boolean AND C-p has no complement for any p not equal 2. However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices L, Alg L boolean AND C-p fails to have a complement for any p not equal 2.