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Comments on the completeness of order complements and on the Prüfer numbers. (English) Zbl 0657.06006

Given a poset E we define two chain extensions of it; the first is the system of all its chains without end points. The second \((E^*\) in symbols) is a system of directed sets for which a linear subset is cofinal therein. \(E^*\) is complete, i.e. \((E^*)^*=E^*\), as well as its maximal linear subsets. The two completions applied to the set of non-negative integers ordered by division, give isomorphic complements. Each of them consists of the underlying set for a system which is closed under the operations of product, g.c.d. and l.c.m. of any number of elements; the set is analogous to the Prüfer group \(p^{\infty}\), for p prime.
Reviewer: J.N.Stabakis

MSC:

06A06 Partial orders, general
06B23 Complete lattices, completions
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References:

[1] Dokas ( L. ).- Certains complétés des ensembles ordonnés munit d’opérations . Complété de KRASNER, C.R.Acad. Paris , t. 256 , 1963 , p. 3937 - 39 . MR 160736 | Zbl 0158.01605 · Zbl 0158.01605
[2] Erne ( M. ). - Posets isomorphic to their extensions , Order , t. 2 , 1985 , p. 199 - 210 . MR 815865 | Zbl 0575.06003 · Zbl 0575.06003
[3] Fuchs ( L. ). - Infinite Abelian Groups , Tomes I , II , Academic Press N. York London 1973 . MR 349869 | Zbl 0209.05503 · Zbl 0209.05503
[4] Redei ( L. ).- Algebra ., Pergamon Press. Oxford , 1967 . Zbl 0191.00502 · Zbl 0191.00502
[5] Stratigopoulos ( D. ), Stabakis ( J. ).- Sur les ensembles ordonnés (f*-coupes) , C.R.Acad. Paris , t. 285 , 1977 , p. 81 - 84 . MR 441733 | Zbl 0362.06006 · Zbl 0362.06006
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