A necessary and sufficient condition for embedding principally decomposable finite lattices into the computably enumerable degrees preserving greatest element

We present a necessary and sufficient condition for embedding principally decomposable finite lattices into the computably enumerable (c.e.) degrees preserving greatest element. In the earlier work Lerman [19] gave a necessary and sufficient condition for embeddings of principally decomposable lattices into the c.e. degrees that do not preserve greatest element. Here, we present the construction of an embedding of a principally decomposable lattice that preserves greatest element, prove that Lerman's condition is sufficient for such an embedding construction and show that the necessity of the condition follows from [19].