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A topological representation of lattices. (English) Zbl 0790.06005

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 a given bounded lattice \(L\) the congruence of \(L\) correspond to certain subcontexts of \(\mathbb{K}^ \tau(L)\); an intrinsic description of these subcontexts is given.
With the aid of this theory, the author proves the (known) theorem that for every distributive bialgebra lattice \(D\) there exists a bounded lattice \(L\) such that \(D\) is isomorphic to the lattice of all congruences of \(L\).
The final part of the paper deals with connections between the preceding theory and known representation theorems for Boolean lattices, distributive bounded lattices and arbitrary bounded lattices.

MSC:

06B15 Representation theory of lattices
06B23 Complete lattices, completions
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References:

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