Algebraic hypermap morphisms
Algebraic hypermaps combinatorially describe topological hypermaps, that is, embeddings of hypergraphs in compact orientable surfaces. For any set \(\Omega\) denote by \(\text{Sym}(\Omega)\) the group of all permutations of \(\Omega\). An algebraic hypermap \(H\) is a triple \((\Omega,\sigma,\alpha)\), where \(\Omega\) is a finite set, \(\sigma,\alpha\in\text{Sym}(\Omega)\), and the group \(\langle\sigma,\alpha\rangle\) generated by \(\sigma\) and \(\alpha\) is transitive on \(\Omega\). In this paper the authors introduce hypermap morphisms, a natural notion of degree and brach number of hypermap morphisms. Theorem 1.1. Let \(\Phi\colon H\to H'\) be a hypermap morphism with degree \(m\) and branch number \(B\). Let \(g\) and \(g'\) be the genera of \(H\) and \(H'\), respectively. Then \(2g-2=m(2g'-2)+B\).
scientific article; zbMATH DE number 1512030