Tensor products of contexts and complete lattices
type: Item
instance of: scholarly article
title: Tensor products of contexts and complete lattices
zbMATH Open document ID: 0799.06009
DOI: 10.1007/BF01188179
author: Marcel Erné
published in: Algebra Universalis
publication date: 1994-11-17T00:00:00Z
review text: Contexts, which play a fundamental role in \textit{R. Wille}'s formal concept analysis [in: Ordered sets, Proc. NATO Adv. Stud. Inst., Banff/Can. 1981, 445-470 (1982; Zbl 0491.06008)], are just binary relations between sets. Although they agree with Boolean Chu spaces as recently studied by Pratt and others, the notion of conceptual morphism between contexts is fundamentally different from the notion of a Chu transform, so the analogy ends right here. The concept lattice of a context essentially is the set of fixed points of the polarity (= Galois connection between the power sets) induced by the relation, suitably ordered. Consequently, the concept lattice of the order relation of a poset is essentially the Dedekind-MacNeille completion of the poset. Conversely, the order relation of a concept lattice may ve viewed as a completion of the original context. This induces a reflection on \textbf{CC}, and the corresponding reflective subcategory may be identified with the category \textbf{CLC} of complete lattices and complete homomorphisms. \textit{R. Wille} introduced a binary operation called tensor product for complete lattices [Order 2, 81-95 (1985; Zbl 0583.06007)], which he later generalized to arbitrarily many factors. Instead of the pointwise order on the cartesian product of the underlying sets, one considers the relation between tuples that requires the elements in at least one component to satisfy the order relation of the corresponding factor, and then one passes to the concept lattice. Besides generalizing this notion from \textbf{CLC} to \textbf{CC}, for the original construction in \textbf{CLC} the author constructs a canonical family of complete homomorphisms from the factors into the tensor product. He then shows that this family has a universal property in \textbf{CLC} just like that for coproducts, however not with respect to arbitrary sinks (= families of complete homomorphisms with common codomain), but with respect to so-called distributing sinks.…
reviewed by: J. Koslowski
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06D10
zbMATH DE Number: 513019
zbMATH Keywords: complete distributivity
zbMATH Keywords: formal concept analysis
zbMATH Keywords: conceptual morphism
zbMATH Keywords: contexts
zbMATH Keywords: concept lattice
zbMATH Keywords: complete lattices
zbMATH Keywords: complete homomorphisms
zbMATH Keywords: tensor product
zbMATH Keywords: universal property
zbMATH Keywords: distributing sinks
zbMATH Keywords: categorical coproduct
MaRDI profile type: Publication
cites work: Tensor Products and Bimorphisms
cites work: Coproducts of bounded \((\alpha,\beta)\)-distributive lattices
cites work: Q3344249
cites work: The Dedekind-MacNeille completion as a reflector
cites work: Distributive laws for concept lattices
cites work: Tensor products for bounded posets revisited
cites work: Q4111633
cites work: The structure of Galois connections
cites work: The tensor product of distributive lattices
cites work: Q3952147
cites work: Tensorial decomposition of concept lattices
cites work: Q3979121
Recommended article: Tensor products of complete lattices and their application in constructing quantales
Recommended article: Q3758893
Recommended article: Some characterizations of the tensor product of complete lattices with applications to quantales
Recommended article: Q3979121
Recommended article: The tensor product of distributive lattices: structural results
Recommended article: Q4726297
Recommended article: Tensor products of \(n\)-complete algebras
Recommended article: Tensor products of reflexive subspace lattices
Recommended article: Q5200279
Recommended article: Sums, products and negations of contexts and complete lattices
description: scientific article; zbMATH DE number 513019