Lattices of algebraic subsets and implicational classes. (English) Zbl 1375.06005

Grätzer, George (ed.) et al., Lattice theory: special topics and applications. Volume 2. Basel: Birkhäuser/Springer (ISBN 978-3-319-44235-8/pbk; 978-3-319-44236-5/ebook). 103-151 (2016).
Chapter IV of the collective monograph gives a broad overview of results obtained in the line of investigation mentioned in the title. Here, the list of contents according to sectional partition is cited. 4-1. Lattices of algebraic subsets and associated subclasses (4-1.1. Lattices of algebraic subsets in power set lattices; 4-1.2. Algebraic subsets of complete lattices; 4-1.3. Congruence semidistributive varieties of algebras; 4-1.4. Perfect lattices).
4-2. Closure systems and implications (4-2.1. Standard and reduced closure systems; 4-2.2. Closure operators and implications).
4-3. Lattices of quasi-equational theories (4-3.1. Subalgebras of lattices of algebraic subsets; 4-3.2. Lattices of quasivarieties versus lattices of quasi-equational theories; 4-3.3. Congruence lattices of semilattices with operators; 4-3.4. Representation theorems; 4-3.5. Atomistic lattices of quasivarieties).
4-4. Exercises.
4-5. Problems (For example, Problem 4.12: Let \(L\) be the lattice consisting of the empty set and all cofinite subsets of an infinite set \(X\). Can \(L\) be represented as the congruence lattice of a semilattice with operators? As a lattice of quasi-equational theories?)
For the entire collection see [Zbl 1357.06001].


06B05 Structure theory of lattices
06B15 Representation theory of lattices
06B20 Varieties of lattices
08C15 Quasivarieties
06A15 Galois correspondences, closure operators (in relation to ordered sets)
Full Text: DOI


[1] pp.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.