The Algebra of Signatures for Extreme Two-Uniform Hypergraphs

In the last decade, several characterizations have been constructed for constructions such as extreme hypergraphs. One of the most recently described features is the signature. A signature is a number that uniquely describes an extremal and allows one to efficiently store the extremal two-uniform hypergraph itself. However, for the signature, although various algorithms have been derived for transforming it into other object-characteristics such as the base, the adjacency matrix, and the vector of vertex degrees, no isolated signature union and intersection apparatus has been constructed. This allows us to build efficient algorithms based on signatures, the most compact representation of extremal two-uniform hypergraphs. The nature of the algebraic construction that can be built on a set of signatures using union and intersection operations has also been defined. It is proved that an algebra on a set of signatures with either the union or intersection operation forms a monoid; if the algebra is defined on a set of signatures with both union and intersection operations, it forms a distributive lattice.