Absolute Graphs with Prescribed Endomorphism Monoid
نویسنده
چکیده
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 End(Γ) and M is absolute, i.e. End(Γ) ∼= 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 [15] into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see [8].
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