On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements
type: Item
instance of: scholarly article
title: On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements
zbMATH Open document ID: 1195.06005
author: Christian Herrmann
published in: Algebra Universalis
publication date: 2009-11-25T00:00:00Z
review text: 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.
reviewed by: Grigore Călugăreanu
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06C20
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 5638425
zbMATH Keywords: perfect element
zbMATH Keywords: modular lattice
zbMATH Keywords: neutral element
zbMATH Keywords: representation of posets
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: preprojective
zbMATH Keywords: preinjective
zbMATH Keywords: GP-system of perfect elements
MaRDI profile type: Publication
full work available at URL: s00012-009-0007-1
OpenAlex ID: W2011467679
cites work: On the elementary theory of quadruples of vector spaces
cites work: Q5532570
cites work: On four subspaces of a vector space
cites work: Q3956998
cites work: Q4288683
cites work: Note on complemented modular lattices
cites work: Perfect elements in the free modular lattices
cites work: Q4071780
cites work: Q4129496
cites work: Q3912633
cites work: Rahmen und erzeugende Quadrupel in modularen Verbänden
cites work: On elementary Arguesian lattices with four generators
cites work: Zum Wortproblem für freie Untermodulverbände
cites work: Q4163274
cites work: Existence varieties of regular rings and complemented modular lattices
cites work: Acyclic modular lattices and their representations
cites work: A test for identities satisfied in lattices of submodules
cites work: Tame algebras and integral quadratic forms
cites work: Hammocks
cites work: Q4271281
cites work: Gelfand-Ponomarev and Herrmann constructions for quadruples and sextuples
cites work: Q3972808
DOI: 10.1007/S00012-009-0007-1
description: scientific article; zbMATH DE number 5638425