Notes on sublattice-lattices
type: Item
instance of: scholarly article
title: Notes on sublattice-lattices
zbMATH Open document ID: 0921.06006
DOI: 10.1023/A:1004509602086
published in: Periodica Mathematica Hungarica
publication date: 1999-09-14T00:00:00Z
review text: For a lattice \(L\), let \(\text{Sub} (L)\) denote the lattice of all sublattices of \(L\), including the empty set. The lattice \(\text{Sub}(L)\) is called the sublattice-lattice of \(L\). The lattice \(\text{Sub} (L)\) determines \(L\) if and only if, for an arbitrary lattice \(K, \text{Sub} (L)\cong \text{Sub}(K)\) implies \(L\cong K\) or \(L\cong K^d\). The author proves the following three results: Theorem 1. If \(L\) is a directly reducible lattice, then \(\text{Sub} (L)\) determines \(L\). Theorem 2. Selfdual subdirectly irreducible lattices of locally finite length are determined by their sublattice-lattices. Corollary 3. Selfdual splitting varieties are closed under the isomorphism of sublattice-lattices. The author also gives a class of selfdual varieties that are not closed under the isomorphism of sublattice-lattices, and observes that the characterization of closed varieties is still open.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 03G10
zbMATH DE Number: 1336271
zbMATH Keywords: lattice
zbMATH Keywords: sublattice-lattice
zbMATH Keywords: direct product
zbMATH Keywords: splitting lattice
zbMATH Keywords: homogeneous prime sublattice
author: Géza Takách
reviewed by: Thomas B. Muenzenberger
MaRDI profile type: Publication
full work available at URL: a:1004509602086
OpenAlex ID: W1900487815
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description: scientific article; zbMATH DE number 1336271