Minor classes

For many classes of combinatorial structures, such as graphs and matroids, there exists a concept of point removal. Minors are obtained by various manners of point removal. In this thesis, these ideas are abstracted to give the definition of minor class. It turns out that minor classes are algebras, in the sense of universal algebra, which makes much universal algebra theory available to the study of minor classes. For example, varieties of minor classes are studied, as well as subalgebras (sub minor classes), homomorphisms, and direct products. Amongst the theory developed, is a natural connection between varieties of minor classes and categories. Also it is shown how a minor class can be described in terms of its so called ˜ìv†-structures and natural excluded minors (which are its excluded minors in the so called completion of the minor class). Many well known minor classes are described in this way, including the minor class of matroids, various minor classes of graphs, and minor classes of subspaces of certain vector spaces over a field (related to Tutte's chain groups). For any field, the latter minor class has, as a homomorphic image, the minor class of matroids coordinatisable over that field. This provides a motivation for further study of minor class homomorphisms.