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Automorphism groups of covering posets and of dense posets

Published online by Cambridge University Press:  20 January 2009

Gerhard Behrendt
Gerhard Behrendt
Affiliation:
Mathematisches InstitutUniversität TübingenAuf der Morgenstelle 10D-7400 Tübingen 1Fed. Rep. Germany
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Given a poset (X, ≦), the covering poset (C(X), ≦) consists of the set C(X) of covering pairs, that is, pairs (a, b)∈X2 with a<b such that there is no c∈X with a<c<b, partially ordered by (a, b)≦(a′, b′) if and only if (a, b) = (a′, b′) or b≦a′. There is a natural homomorphism v from the automorphism group of (X, ≦) into the automorphism group of (C(X), ≦). It is shown that given groups G, H and a homomorphism α from G into H there exists a poset (X, ≦) and isomorphisms φψ from G onto Aut(X, ≦), respectively from H onto Aut(C(X), ≦) such that φv = αψ. It is also shown that every group is isomorphic to the automorphism group of a poset all of whose maximal chains are isomorphic to the nationals.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 1 , February 1992 , pp. 115 - 120
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

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