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Semigroup Forum

, Volume 76, Issue 2, pp 256–267 | Cite as

Absolute graphs with prescribed endomorphism monoid

  • Manfred Droste
  • Rüdiger Göbel
  • Sebastian Pokutta
Research article

Abstract

We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid \(\mathop{\rm End}\nolimits (\Gamma)\) and M is absolute, i.e. \(\mathop{\rm End}\nolimits (\Gamma)\cong M\) holds in any generic extension of the given universe of set theory. This is true if and only if |M|,|Γ| are smaller than the first Erdős cardinal (which is known to be strongly inaccessible). We will encode Shelah’s absolutely rigid family of trees (Isr. J. Math. 42(3), 177–226, 1982) into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see Göbel and Pokutta (Shelah’s absolutely rigid trees and absolutely rigid fields, in preparation).

Keywords

Absolute endomorphism monoids Action of semigroups on sets Colored graphs Absoluteness 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Manfred Droste
    • 1
  • Rüdiger Göbel
    • 2
  • Sebastian Pokutta
    • 3
  1. 1.Institute of Computer ScienceUniversity of LeipzigLeipzigGermany
  2. 2.Department of MathematicsUniversity of Duisburg-EssenEssenGermany
  3. 3.Operations Research CenterMassachusetts Institute of TechnologyCambridgeUSA

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