## 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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