On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements

Gel'fand and Ponomarev [11] introduced the concept of perfect elements and constructed such in the free modular lattice on 4 generators. We present an alternative construction of such elements u (linearly equivalent to theirs) and for each u a direct decomposition u, u of the generating quadruple within the free complemented modular lattice on 4 generators; u, u are said to form a perfect pair. This builds on [17] and fills a gap left there. We also discuss various notions of perfect elements and relate them to preprojective and preinjective representations.