A topological representation of lattices
type: Item
instance of: scholarly article
title: A topological representation of lattices
zbMATH Open document ID: 0790.06005
DOI: 10.1007/BF01190610
author: Gerd Hartung
published in: Algebra Universalis
publication date: 1992-09-27T00:00:00Z
review text: A context is a triple \(\mathbb{K}= (G,M,I)\), where \(G\) and \(M\) are sets and \(I\) is a relation between \(G\) and \(M\). Given such a context \(\mathbb{K}= (G,M,I)\), for \(A\subseteq G\) and \(B\subseteq M\) one defines \(A':=\{m\in M: gIm\) for all \(g\in A\}\) and \(B':=\{g\in G: gIm\) for all \(m\in B\}\). A concept is a pair \((A,B)\) with \(A\subseteq G\), \(B\subseteq M\), \(A'=B\), \(B'=A\). Under the partial order \((A,B)\leq (C,D)\Leftrightarrow A\subseteq C\), the concepts form a complete lattice \({\mathfrak B}(\mathbb{K})\). Every complete lattice is shown to be isomorphic to such a concept lattice. A topological context is a context \(\mathbb{K}^ \tau=(G,M,I)\), where \(G\) and \(M\) are topological spaces with the properties (1) if \(A\subseteq G\) and \(B\subseteq M\) are closed, then so are \(A'\) and \(B'\); and (2) \(\{A: A= A''\) is closed in \(G\), \(A'\) is closed in \(M\}\) is a subbase for the closed sets in \(G\) and, dually, \(\{B: B= B''\) is closed in \(M,B'\) is closed in \(G\}\) is a subbase for the closed sets in \(M\). For such a topological context \(\mathbb{K}^ \tau= (G,M,I)\) a concept \((A,B)\) is said to be closed if both \(A\) and \(B\) are closed. The closed concepts form a lattice \({\mathfrak B}^ \tau(\mathbb{K}^ \tau)\) that is bounded but not necessarily complete. The paper concerns representation of arbitrary bounded lattices as closed concept lattices. The main results are the following. A topological context satisfying certain minimality conditions is called ``standard''. It is proved that for every bounded lattice \(L\) there exists an essentially unique standard topological contexts \(\mathbb{K}^ \tau(L)\) such that \(L\) is isomorphic to \({\mathfrak B}^ \tau(\mathbb{K}^ \tau(L))\). This representation theorem leads to a dual equivalence between the category of all bounded lattices with surjective homomorphisms and the category of all standard topological context with so-called ``standard'' embeddings. Thus, for…
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 06B23
zbMATH DE Number: 64763
zbMATH Keywords: categorical equivalence
zbMATH Keywords: representation of bounded lattices as closed concept lattices
zbMATH Keywords: lattice of congruences
zbMATH Keywords: topological context
zbMATH Keywords: distributive bialgebra lattice
zbMATH Keywords: representation theorems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 64763