On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements
In 1974, Gel'fand and Ponomarev introduced perfect elements and constructed such elements in the free modular lattice on 4 generators. In the paper under review, among other things, an alternative construction of such elements is presented. Various notions of perfect elements are discussed and related to preprojective and preinjective representations. This difficult paper is divided into 14 sections, some of which are expository. We will just browse some of the results. 1. Introduction. 2. Basic concepts. 3. Auxiliary results. 4. Perfect pairs. 5. Relevant elements of \(F(4)\): here, by simultaneous recursion, some elements \(s_{n},t_{n},p_{ni},r_{ni}\) (neutral for \(n\leq 2\)) are defined. 6. Perfect pairs for quadruples: Theorem 6.1. Each of the elements \(s_{n},t_{n},r_{ni}\) of \(F(4)\) can be completed to a perfect pair. In particular, each is neutral in \(F(4)\). The proof of this Theorem is performed in Section 7 (Perfect pairs with \(s_{n}\)) and Section 8 (Perfect pairs with \(r_{ni}\)). 9. Some theory of modular lattices. 10. Perfect elements. 11. Dependence on the base field. 12. Completeness of systems of perfect elements: Theorem 12.1. For a poset \(S\) admitting a GP-system of perfect elements and a neutral element \(u\) of \( FL_{p}(S)\), \(p<\infty \), either \(u\in B_{p}(S)\) or \(w<u<v\) for all \(w\in B_{p}^{-}(S)\) and \(v\in B_{p}^{+}(S)\). 13. Results on quadruples: Theorem 13.1 has the following Corollary 13.2. For a quadruple \(S\), and within \( FL_{p}(S)\), the concepts of neutral, perfect, and strongly perfect elements are equivalent. For \(p<\infty \), the image of \(B(S)\) under the canonical homomorphism is the sublattice of all neutral elements of \(FL_{p}(S)\). 14. Résumé: Suggestions for further work are given and some conjectures are presented.
scientific article; zbMATH DE number 5638425
scientific article