Compact generation in partially ordered sets

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Erné, Marcel: Compact generation in partially ordered sets. In: Journal of the Australian Mathematical Society 42 (1987), Nr. 1, S. 69-83. DOI: https://doi.org/10.1017/S1446788700033966

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Abstract: 
Several` “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices. © 1987, Australian Mathematical Society. All rights reserved.
License of this version: Es gilt deutsches Urheberrecht. Das Dokument darf zum eigenen Gebrauch kostenfrei genutzt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. Dieser Beitrag ist aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
Document Type: article
Publishing status: publishedVersion
Issue Date: 1987
Appears in Collections:Fakultät für Mathematik und Physik

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1 image of flag of Germany Germany 19 70.37%
2 image of flag of Romania Romania 2 7.41%
3 image of flag of United States United States 1 3.70%
4 image of flag of Taiwan Taiwan 1 3.70%
5 image of flag of Turkey Turkey 1 3.70%
6 image of flag of Serbia Serbia 1 3.70%
7 image of flag of France France 1 3.70%
8 image of flag of Egypt Egypt 1 3.70%

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