A perspective on algebraic representations of lattices
type: Item
instance of: scholarly article
title: A perspective on algebraic representations of lattices
zbMATH Open document ID: 0804.06008
DOI: 10.1007/BF01221791
author: William A. Lampe
published in: Algebra Universalis
publication date: 1994-07-18T00:00:00Z
review text: This is a comprehensive survey of known results on algebraic representations of lattices. After a short historical overview of classical results by Birkhoff, Frink and Whitman, it continues with various attempts at the representation problem. At first, it presents the classical Birkhoff-Frink theorem giving the equivalence of algebraicity of a lattice \(L\) and its representability as \(\text{Sub } A\) for an algebra \(A\) with countable similarity type, which is equivalent to \(L\cong \text{Sub } A\) for a groupoid \(A\) or a commutative Moufang loop. It states the problem of characterizing \(\text{Sub } A\) for a semigroup \(A\). Next, he discuss known results on the representation of \(L\) as a sublattice of \(\text{Eq} (A)\). He quotes the theorems by B. Jónsson: Every \(L\) is isomorphic to a type-3 sublattice of some \(\text{Eq} (A)\). \(L\) has a type-2 partition-lattice representation iff \(L\) is modular. The case of type-1 representation and the property ``to be Arguesian'' is discussed in full detail. The following part is devoted to the famous Grätzer-Schmidt theorem: \(L\) is algebraic iff \(L\cong \text{Con } A\) for some algebra \(A\) (which can be of type 3). Pudlák's proof is considered and his construction is compared with that of the original one by Grätzer and Schmidt. Grätzer and the author proved that if \(L\) is a modular and algebraic lattice, then \(L\cong \text{Con } A\) with \(\text{Con } A\) a type-2 sublattice of \(\text{Eq} (A)\). It is a problem if this can be extended also for a type-1 representation. The next part contains results on the representation of a finite lattice \(L\) as \(\text{Con } A\) for a finite algebra \(A\). The paper lists results by Pálfy, Pudlák, Tůma and McKenzie. The author lists also known results on representations \(L\cong \text{Con } A\) for \(A\) of finite (or countable) similarity type. For uncountable similarity type, he mentions \(k\)-algebraicity of \(L\) for some cardinal \(k\). Although the theorem b…
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 06-02
Mathematics Subject Classification ID: 08-02
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 590885
zbMATH Keywords: algebraic lattice
zbMATH Keywords: subalgebra lattices
zbMATH Keywords: survey
zbMATH Keywords: algebraic representations of lattices
zbMATH Keywords: congruence lattices
MaRDI profile type: Publication
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A perspective on algebraic representations of lattices (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf01221791

A perspective on algebraic representations of lattices (OpenAlex ID)
OpenAlex ID: W1969575024