Abstract
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, \({\bar{u}}\) of the generating quadruple within the free complemented modular lattice on 4 generators; u, \({\bar{u}}\) 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.
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Herrmann, C. On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements. Algebra Univers. 61, 1 (2009). https://doi.org/10.1007/s00012-009-0007-1
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DOI: https://doi.org/10.1007/s00012-009-0007-1