Bigeneration in complete lattices and principal separation in ordered sets. (English) Zbl 0738.06005

Erné, Marcel

Order 8, No. 2, 197-221 (1991).

Summary: By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called \({\mathcal Z}\)-generators resp. \({\mathcal Z}\)- subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) \({\mathcal Z}\)-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable \({\mathcal Z}\), the complete lattices with a smallest join-dense \({\mathcal Z}\)-subbasis consisting of \({\mathcal Z}\)-primes are the \({\mathcal Z}\)-completions of principally separated posets.

MSC:

06B23	Complete lattices, completions
54D15	Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
06B15	Representation theory of lattices
06D05	Structure and representation theory of distributive lattices

Keywords:

join-irreducible; bigeneration; Z-generators; Z-subbases; principal separation; topological separation axiom; complete lattice; join-dense; Z-completions; principally separated posets

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