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A remark on complete lattices represented by sets. (Russian. English summary) Zbl 0135.03202

English summary: I. V. Stelletskiĭ presented necessary and sufficient conditions for a complete lattice \(L\) to be representable by sets (in the sense described in his paper [Usp. Mat. Nauk 12, No. 6(78), 177–180 (1957; Zbl 0081.02403)]). In the present paper it is proved that Stelletskiĭ’s conditions may be formulated in a form which shows a connection of these lattices with compactly generated lattices (see [P. Crawley, Bull. Am. Math. Soc. 65, 377–379 (1959; Zbl 0094.01702)]). An element \(a\) of a complete lattice \(L\) is called chain-compact if for each chain \(\{a_\alpha\}\), \(\alpha\in A\) in \(L\) with \(\displaystyle a\le \bigvee_{\alpha\in A} a_\alpha\) there exists an \(\alpha_0\in A\) such that \(a\le a_{\alpha_0}\). A lattice \(L\) is chain-compactly generated if each of its elements is a join of chain-compact elements. Then the following theorem holds:
\(L\) can be represented by sets if and only if \(L\) is chain-compactly generated.
Theorem 2 characterizes compactly generated lattices by means of the concept of representation by sets. Theorem 3 shows a simple structure of complete lattices which may be represented by sets: In a complete lattice \(L\) representable by sets, if each two quotients \(a\vee b/a\), \(b/a\vee b\) \((a,b\in l)\) are isomorphic, then \(L\) is modular.

MSC:

06B23 Complete lattices, completions
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