## 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

### Keywords:

complete lattices

### Citations:

Zbl 0081.02403; Zbl 0094.01702
Full Text: