Frucht theorem for inverse semigroups
The main result is the following: For any finite inverse semigroup \(S\) there exists a weighted graph \(\Gamma\) such that \(\text{PAut}_\omega\Gamma\cong S\). Here a graph is undirected without loops with vertex set \(V(\Gamma)\) and edge set \(E(\Gamma)\) and weight function \(\omega\colon V(\Gamma)\to P\) where \(P\) is a lower semilattice. \(\text{PAut}_\omega\) denotes the set of partial automorphisms of \(\Gamma\), i.e., partial injective mappings which preserve and reflect edges of \(\Gamma\) with the additional property that \(\omega(a)=\omega(b)\) for a partial automorphism \(\varphi\) implies \(\omega(\varphi(a))=\omega(\varphi(b))\) and \(\omega(a)<\omega(b)\) implies \(\omega(\varphi(a))<\omega(\varphi(b))\) if \(\varphi(a)\) and \(\varphi(b)\) are defined.
scientific article; zbMATH DE number 1238012