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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].

MSC:

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)
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