## 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
Full Text:

### References:

 [1] Aigner, M.,Combinatorial theory. Springer-Verlag, Berlin/Heideiberg/New York, 1919. · Zbl 0858.05001 [2] Balbes, R. andDwinger, P.,Distributive lattices. Univ. of Missouri Press, Columbia, 1974. · Zbl 0321.06012 [3] Barwise, J. (ed.),Handbook of mathematical logic. North-Holland Publishing Company, Amsterdam/New York/Oxford, 1977. [4] Birkhoff, G.,Lattice theory Third edition. American Mathematical Society, Providence, R.I., 1967. · Zbl 0153.02501 [5] Crawley, P. andDilworth, R. P.,Algebraic theory of lattices. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. [6] Erné, M.,Einführung in die Ordnungstheorie. B.I.-Wissenschaftsverlag, Mannheim/Wien/Zürich, 1982. [7] Erné, M.,Topologie. Ausarbeitung einer im Sommersemester 1984 gehaltenen Vorlesung. Hannover, 1984. [8] Ganter, B.andWille, R.,Skriptum zur Vorlesung ?Formale Begriffsanalyse?. Darmstadt, 1987. [9] Gierz, G., Hofmann, H., Keimel, K., Lawson, J. D., Mislove, M. andScott, D. S.,A compendium of continuous lattices. Springer-Verlag, Berlin/Heidelberg/New York, 1980. · Zbl 0452.06001 [10] Gumm, H. P. andPoguntke, W.,Boolesche Algebra. B.I.-Wissenschaftsverlag, Mannheim/Wien/Zürich, 1981. [11] Hartung, G.,Darstellung beschränkter Verbände als Verbände offener Begriffe. Diplomarbeit. Darmstadt, 1989. [12] Herrlich, H. andStrecker, G. E.,Category theory. Allyn and Bacon Inc., Boston, 1973. [13] Johnstone, P. T.,Stone spaces. Cambridge University Press, Cambridge/New York/Melbourne, 1982. · Zbl 0499.54001 [14] Kelley, J. L.,General topology. D. Van Nostrand Company, Inc., Princeton, N.J./Toronto/London/Melbourne, 1968. [15] Priestley, H. A.,Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc.,2 (1970), 186-190. · Zbl 0201.01802 [16] Reuter, K. andWille, R.,Complete congruence relations of concept lattices. Acta Sic. Math.,51 (1987), 319-327. · Zbl 0668.06005 [17] Stahl, J. andWille, R.,Preconcepts and set representation of contexts. In: W. Gaul and M. Schader (eds.): Classification as a tool of research. Elsevier Science Publishers B. V. (North-Holland), Amsterdam, 1986, 431-438. [18] Stone, M. H.,Topological representations of distributive lattices and Brouwerian logics. ?asopis p?st, mat.,67 (1937), 1-25. · Zbl 0018.00303 [19] Urquhart, A.,A topological representation theory for lattices. Algebra Universalis,8 (1978), 45-58. · Zbl 0382.06010 [20] Wille, R.,Restructuring lattice theory: an approach based on hierarchies of concepts. In: I. Rival (ed.): Ordered sets. Reidel, Dordrecht/Boston, 1982, 445-470. · Zbl 0491.06008 [21] Wille, R.,Subdirect decomposition of concept lattices. Algebra Universalis,17 (1983), 275-287. · Zbl 0539.06006 [22] Wille, R.,Complete tolerance relations of concept lattices. In: G. Eigenthaler, H. K. Kaiser, W. B. Müller and W. Nöbauer (eds.): Contributions to general algebra3. Hölder-Pichler-Tempsky, Wien, 1985, 397-415. · Zbl 0563.06006 [23] Wille, R.,Tensorial decomposition of concept lattices. ORDER2 (1985), 81-95. · Zbl 0583.06007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.