Endomorphism semigroups for weak \(p\)-hypergraphs
A hypergraph \((V,E)\) is called a weak \(p\)-hypergraph if it satisfies: (a) for any \(v\in V\) there exists \(A\in E\) with \(v\in A\); (b) \(|A|\geq p+1\) for all \(A\in E\) and for every \(p\)-element subset \(B\subseteq V\) there exists at most one \(A\in E\) with \(B\subseteq A\); (c) there exist \(p+1\) vertices of \(V\) contained in no \(A\in E\). The endomorphism monoids of weak \(p\)-hypergraphs are described by formulas in the elementary theory of semigroups. Two distinct weak \(p\)-hypergraphs have distinct endomorphism monoids and two weak \(p\)-hypergraphs are isomorphic if and only if their endomorphism monoids are isomorphic. Consequences for affine and projective geometries are formulated. The proofs are omitted.
scientific article; zbMATH DE number 1541587